Introduction to Algebra
Algebra · 102 lessons
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Algebra is defined as **“the art of bringing together unknowns (or unknown quantities) to match a known quantity.”** A **symbol or quantity** that represents **one and only one value** throughout a particular discussion is called a **constant**, while a symbol that may represent **any one of a collection of values** is called a **variable**. Any meaningful **collection of constants, variables and operations**, such as $4x + 7y - z$, is called an **algebraic expression**, or simply an expression. The **terms** of an expression are those quantities related by addition and subtraction. $$Example: 4x, 7y, z are the terms in 4x + 7y - z$$ The **factors** of an expression are those quantities related by multiplication. $$Example: The term 4x has factors 4 and x $$ ## Ohm’s Law quote The famous **Ohm’s Law** in electrical and electronics engineering can be written in **three algebraic equations or algebraic formula**: center [width=250px,height=250px]{https://dr7ru8o7bbhub.cloudfront.net/DEV/MATH/ITE/Algebra_1_new2.png} $$V = IR, I = V{R}, R = V{I}$$ center Instead of specifying the voltage, current, and resistance values in numbers, we use symbols like $V$, $I$, and $R$ to denote generalised quantities. - $V = IR$ and $R = V{I}$ are examples of **algebraic expressions** - $I = V{R}$ and others are **algebraic equations** or **formula** Frequently, it is convenient to **transform** or **transpose** an algebraic formula or equation to express it with a **different subject**. quote ## Rules of BODMAS quote The rules of **BODMAS** are similar to the laws of algebra. BODMAS stands for: - **B: Brackets** – Do what is inside the brackets first, starting from the innermost brackets and moving outward. - **O: Orders** – Follow the orders in mathematics (i.e., powers or roots). - **DM: Division and Multiplication** – Perform divisions and multiplications from left to right. - **AS: Addition and Subtraction** – Finally, perform additions and subtractions from left to right. **Example 1:** $$Simplify -3 + 5 (7^2 + 5)$$ **Step 1:** Solve the parentheses $(7^2 + 5)$ Follow the order of operations: exponents before addition $$7^2= 77$$ $$=49$$ $$49+5 = 54 $$ **Step 2:** Rewrite the expressions $$-3 + 5 (7^2+5) $$ $$becomes:$$ $$-3 + 554$$ **Step 3:** Perform multiplication before addition $$554=270$$ **Step 4:** Perform the final addition $$-3 + 270 = 267$$ **Final Answer:** 267 **Example 2:** $$Simplify 12(5 + 6 3) (4 + 2)$$ **Step 1:** Solve the first brackets $(5+63)$ Follow the order of operations: multiplication before addition $$63=18$$ $$5+18=23$$ **Step 2:** Solve the second brackets $4+2$ $$4+2 = 6$$ **Step 3:** Rewrite the expression $$12(5+63) (4+2)$$ $$becomes:$$ $$12236$$ **Step 4:** Perform multiplication and division from left to right $$First do: 1223=276$$ $$Then do: 276 6 = 46$$ **Final Answer:** 46 quote
Introduction to Algebra
Simplify the expression: $ 9 - 2 \times (5^2 - 1) $
Correct answer: A. -39
• Why A is correct: Follow order of operations (PEMDAS). First, solve the exponent: 5² = 25. Then inside the parentheses: 25 - 1 = 24. Next, multiply: 2 × 24 = 48. Finally, subtract: 9 - 48 = -39.
• Why B is wrong (-1): This results from incorrectly calculating 2 × (5² - 1) as 10 instead of 48, suggesting you may have skipped the exponent or made an arithmetic error.
• Why C is wrong (1): This happens if you subtract before multiplying, violating order of operations (doing 9 - 2 = 7, then attempting other calculations).
• Why D is wrong (-41): This is close but suggests an arithmetic mistake in the final subtraction, possibly calculating 9 - 50 instead of 9 - 48.
If $ I = 0.5 $ A and $ R = 20 \, \Omega $, calculate $ V $ using $ V = IR $
A. 10 ✓
Using the formula V = IR, multiply 0.5 A × 20 Ω = 10 V. This is straightforward application of Ohm's Law.
B. 40 — This would result from multiplying 2 × 20, using the wrong current value.
C. 0.025 — This comes from dividing instead of multiplying (0.5 ÷ 20), reversing the formula.
D. 100 — This would be correct if R were 200 Ω instead of 20 Ω, or if you used 5 A instead of 0.5 A.
Which of the following is a constant in the expression $ 6x + 2y - 9 $?
Why -9 is correct:
A constant is a number that doesn't change—it has no variable attached to it. In this expression, -9 stands alone, so it's the constant.
Why the others are wrong:
- x and B. 2y: These are variables or terms with variables, so they change depending on what values x and y take—they're not constant.
- D. 6: This is a coefficient (a number multiplying a variable), not a constant term. The 6 is attached to x, so it's part of a variable term, not independent.
Simplify the expression: $ 3 + 6 \times (2^2 + 1) $
Why A (33) is correct:
Follow the order of operations (PEMDAS): First solve the parentheses: 2² + 1 = 4 + 1 = 5. Then multiply: 6 × 5 = 30. Finally add: 3 + 30 = 33.
Why the others are wrong:
- B (21): Ignores the exponent and just does 6 × (2 + 1) = 18, then 3 + 18 = 21.
- C (15): Incorrectly adds before multiplying: treats it as (3 + 6) × (2 + 1) = 9 × 5 = 45, then somehow gets 15 (calculation error).
- D (45): Solves (3 + 6) × (2² + 1) = 9 × 5, ignoring that only the terms in parentheses should be grouped, not the 3.
Which of the following is an algebraic equation?
A is correct: An algebraic equation must have an equals sign (=) showing two expressions are equal, and contain a variable (like x). "x + 3 = 10" does both.
B is wrong: This is an algebraic *expression*, not an equation—it has no equals sign, so nothing is being set equal to anything.
C is wrong: Also just an expression. It has variables but no equals sign, so it's not an equation.
D is wrong: This is just arithmetic (4 + 5 × 2 = 14). It has no variable and no equals sign in the proper form of an equation.
Simplify the expression: $ 8 + 3 \times (2^2 + 6) $
Why A (38) is correct:
Follow order of operations (PEMDAS): First solve the parentheses: 2² + 6 = 4 + 6 = 10. Then multiply: 3 × 10 = 30. Finally add: 8 + 30 = 38.
Why the others are wrong:
- B (26): You might get this by forgetting to multiply 3 × 10, just doing 8 + 10 + 6 instead.
- C (22): This happens if you add before multiplying (ignoring order of operations), like 8 + 3 = 11, then 11 × 2 = 22.
- D (36): You might get this by calculating 3 × (2² + 6) = 30, then adding 6 again by mistake.
If $ V = 18 $ volts and $ I = 3 $ A, find resistance $ R $ using $ R = \frac{V}{I} $
Correct answer: A (6)
Using the formula R = V/I, substitute the values: R = 18/3 = 6 ohms. This is correct.
Why the others are wrong:
- B (54): This is what you'd get if you *multiplied* V × I instead of dividing—a common mistake.
- C (15): This comes from subtracting V − I = 18 − 3, not the right operation.
- D (5): This doesn't match any correct calculation with these numbers.
Which is not a term in the expression $ 5x + 4y - 2 $?
Why A is correct:
A term is a single number or variable, or a number multiplied by variable(s). The expression 5x + 4y - 2 has three terms: 5x, 4y, and -2. Since xy doesn't appear in this expression, it's not a term.
Why the others are wrong:
- B (5x) is a term—it's the first part of the expression
- C (4y) is a term—it's the second part of the expression
- D (-2) is a term—it's the constant (final part) of the expression
Rearrange $ R = \frac{V}{I} $ to make $ V $ the subject.
Why A is correct:
To isolate V, multiply both sides of R = V/I by I. This cancels the I on the right side, giving you V = IR.
Why the others are wrong:
- B (V = 1/IR): This comes from incorrectly flipping the fraction instead of multiplying both sides.
- C (V = I/R): This reverses the relationship—you'd get this if you rearranged for the wrong variable.
- D (V = R - I): This uses subtraction, but the original formula uses division, so you need multiplication to undo it.
Which of these is a variable in the expression $ 3a^2 + 5b - 1 $?
A. a ✓
A variable is a symbol that represents an unknown number that can change. In this expression, a and b are variables because they stand for unknown values.
Why the others are wrong:
- B (3) and C (5) are coefficients—numbers that multiply the variables, not the variables themselves
- D (1) is a constant—a fixed number that doesn't change
Simplify the expression: $ 4 + 5 \times (3^2 - 2) $
A. 39 is correct — Follow order of operations (PEMDAS): First, solve the parentheses: 3² = 9, then 9 - 2 = 7. Next, multiply: 5 × 7 = 35. Finally, add: 4 + 35 = 39.
B. 29 — This results from incorrectly calculating 3² - 2 as 5, then doing 5 × 5 = 25, then 4 + 25 = 29. You likely forgot that 3² = 9, not 3.
C. 31 — This comes from adding before multiplying: (4 + 5) × (3² - 2) = 9 × 7 = 63... no, this would give 63. This might result from a different miscalculation of the steps.
D. 19 — This happens if you skip the exponent and treat 3 - 2 = 1, then 5 × 1 = 5, then 4 + 5 = 9... or other operation errors. You'd need to ignore the exponent entirely.
If $ V = 15 $ volts and $ R = 5 \, \Omega $, find current $ I $ using $ I = \frac{V}{R} $
Correct Answer: A (3)
Using the formula I = V/R, substitute the values: I = 15/5 = 3 amps. This is the direct calculation.
Why others are wrong:
- B (10): This would result from 15 − 5, subtracting instead of dividing.
- C (20): This comes from adding 15 + 5, the wrong operation entirely.
- D (75): This results from multiplying 15 × 5, which reverses the formula (that would give you V given I and R, not I).
Which of the following is not a factor of $ 12xy $?
Why A is correct:
A factor must divide evenly into an expression. While 12, x, and y all divide into 12xy perfectly, (x + y) does not—you can't simplify 12xy by pulling out (x + y) as a common factor.
Why the others are wrong:
- B (12): 12 is literally part of the term 12xy, so it's definitely a factor
- C (x): x divides evenly into 12xy, leaving 12y
- D (y): y divides evenly into 12xy, leaving 12x
Rearrange $ I = \frac{V}{R} $ to make $ R $ the subject.
Why A is correct:
To isolate R, multiply both sides by R, then divide both sides by I. This gives you R = V/I.
Why the others are wrong:
- B: Flips the fraction—this would give you I/V instead of V/I.
- C: Uses subtraction instead of division; the original formula has division, not subtraction.
- D: This just rearranges the original equation without solving for R; R still appears on both sides.
Which of the following is an algebraic expression?
Correct Answer: 7a + 4b - 2
An algebraic expression combines variables (letters), numbers, and operations (+, −, ×, ÷) without an equals sign or inequality symbol.
Why the others are wrong:
- 3x = 12: This is an *equation*, not an expression—it has an equals sign
- x < 5: This is an *inequality*, not an expression—it uses a comparison symbol (<)
- 8 + 6 = 14: This is a *number sentence* or equation—it has an equals sign and no variables
Simplify the expression: $ 10 + 4 \times (2^3 - 2) $
Why A (34) is correct:
Follow the order of operations (PEMDAS): First solve the exponent 2³ = 8, then subtract inside the parentheses: 8 - 2 = 6. Next multiply: 4 × 6 = 24. Finally add: 10 + 24 = 34.
Why the others are wrong:
- B (30): You'd get this if you forgot to calculate the exponent and just did 4 × (2 - 2) = 0, then 10 + 0 + 20—a mix-up of operations.
- C (26): This comes from incorrectly computing 2³ as 6 instead of 8, giving 4 × 4 = 16, then 10 + 16 = 26.
- D (40): This results from adding before multiplying—treating it as (10 + 4) × (8 - 2) = 14 × 6—which violates order of operations.
If $ V = 42 $ volts and $ I = 7 $ A, calculate resistance $ R $ using $ R = \frac{V}{I} $
Why A is correct:
Using the formula R = V/I, substitute the values: R = 42/7 = 6 ohms. Division is the correct operation here.
Why the others are wrong:
- B (49): This is V + I (42 + 7), not division—wrong operation.
- C (35): This is V − I (42 − 7), subtraction instead of division.
- D (7): This is just the current value I; you didn't apply the formula at all.
Which one is a term in the expression $ 9p - 2q + 5 $?
-2q is correct because a term is a single part of an expression separated by + or − signs. In 9p - 2q + 5, the three terms are 9p, -2q, and 5 (including the negative sign with -2q).
Why the others are wrong:
- pq — This doesn't appear anywhere in the expression; there's no multiplication of p and q.
- 2p — While p appears in the expression, the term is 9p, not 2p.
- q² — q is in the expression, but it's not squared; it's just q with coefficient -2.
Make $ I $ the subject in the equation $ V = IR $
Correct Answer: A
To make I the subject, you need to isolate it on one side. Since V = IR, divide both sides by R to get I = V/R.
Why the others are wrong:
- B (I = VR): This multiplies V and R, which is the opposite of what we need—we should divide, not multiply.
- C (I = V - R): Subtraction doesn't appear in the original equation. We rearrange using division, not subtraction.
- D (I = R/V): This flips the fraction upside down. We divide V by R, not R by V.
Which of the following is a variable in $ 6x + 9y - 3 $?
Why A is correct: A variable is a letter that represents an unknown number that can change. In this expression, y is a letter, making it a variable.
Why the others are wrong:
- B (6): This is a coefficient—a number that multiplies the variable x, not a variable itself.
- C (3): This is a constant—a fixed number with no letter attached.
- D (9): This is also a coefficient that multiplies y, not a variable.
If $ V = 12 $ volts and $ R = 3 \, \Omega $, what is the current $ I $?
Correct answer: A (4)
Use Ohm's Law: I = V ÷ R = 12 ÷ 3 = 4 amps.
Why the others are wrong:
- B (3): This is just the resistance value; you divided wrong or confused the formula.
- C (36): You multiplied V × R instead of dividing them.
- D (0.25): You flipped the formula (R ÷ V instead of V ÷ R).
Simplify the expression: $ 10 + 3 \times (4^2 - 6) $
Why A (40) is correct:
Follow order of operations (PEMDAS): First do the exponent 4² = 16, then subtract inside parentheses: 16 - 6 = 10, then multiply: 3 × 10 = 30, then add: 10 + 30 = 40.
Why the others are wrong:
- B (28): Likely skipped the exponent and calculated 3 × (4 - 6) = -6, then 10 + 3 × (-6) = -8 (or similar error)
- C (46): Probably did 3 × 4² = 48 without subtracting the 6 first, then added 10
- D (52): Likely multiplied before handling parentheses correctly, or added all numbers without proper order of operations
Which of these is a constant in $ 8m + 2n - 7 $?
• -7 is correct because a constant is a number that doesn't change—it has no variable attached to it. The -7 stands alone in the expression.
• m is wrong because it's a variable (it can change value), not a constant.
• 2n is wrong because it contains a variable (n), so it's not constant.
• 8 is wrong because although 8 is a number, it's a *coefficient* (it's attached to the variable m), not a standalone constant.
Which of the following is an algebraic equation?
Why A is correct:
An algebraic equation has a variable, numbers, and an equals sign (=). "r + 5 = 10" has all three, so it's an equation you can solve.
Why the others are wrong:
- B (6p + 2q): This is an expression, not an equation—there's no equals sign, so nothing to solve.
- C (t - 4 < 6): This is an inequality (uses <), not an equation. Equations use =.
- D (3 × 7): This is just arithmetic with no variable or equals sign—not an algebraic equation.
Simplify: $ 4 + 3 \times \left[2^3 + 4(3 - 1)\right] \div 2 $
Correct answer: A (28)
Follow the order of operations (PEMDAS):
- Parentheses first: 3 − 1 = 2, then 4 × 2 = 8
- Exponents: 2³ = 8
- Inside brackets: 8 + 8 = 16
- Division: 16 ÷ 2 = 8
- Multiplication: 3 × 8 = 24
- Addition: 4 + 24 = 28 ✓
Why others are wrong:
- B (40): Ignores the division by 2; treats it as 4 + 3 × 16 instead of 4 + 3 × 8
- C (32): Likely skipped the division step entirely
- D (26): Probably made an arithmetic error in the brackets or final steps
Simplify using BODMAS: $ \frac{36}{3^2} + 4 \times 2 $
Correct answer: A (12)
• BODMAS means we do Brackets, Orders (powers), Division/Multiplication, then Addition/Subtraction.
• First, solve the power: 3² = 9
• Then divide: 36 ÷ 9 = 4
• Then multiply: 4 × 2 = 8
• Finally add: 4 + 8 = 12 ✓
Why others are wrong:
- B (20): Likely added 36 + 4 first, ignoring order of operations
- C (10): Probably made an error in the division or addition step
- D (16): Likely forgot to divide 36 by 9, or miscalculated the power
What are the terms in the expression $ 6x^2 - 4xy + 3 $?
Why A is correct:
Terms are the individual parts of an expression separated by + or − signs. In 6x² - 4xy + 3, the three terms are 6x², -4xy (including the negative sign), and 3.
Why the others are wrong:
- B: Missing the constant term 3, and wrote 4xy instead of -4xy (the sign matters).
- C: These are the variables and powers, not the terms—you need the coefficients (6, -4) included.
- D: These are just the coefficients and constants, not complete terms; missing the variables and exponents.
Given the expression $ 5x + 3y - 7z $, what are the factors of the term $ 3y $?
Correct answer: A (3 and y)
A term's factors are the numbers and variables that multiply together to make that term. The term 3y is made by multiplying 3 × y, so its factors are 3 and y.
Why the others are wrong:
- B: 5 is part of a different term (5x), not part of 3y
- C: z doesn't appear in the term 3y at all
- D: z isn't in this term; only y is the variable factor
Simplify: $ 2^4 + 3 \times (10 - 4^2 \div 2) $
Correct answer: 22
• Work inside parentheses first: 4^2 ÷ 2 = 16 ÷ 2 = 8
• Then: 10 - 8 = 2
• Multiply: 3 × 2 = 6
• Add: 2^4 + 6 = 16 + 6 = 22 ✓
Why others are wrong:
- B (64): You'd get this if you ignored the division and subtraction, treating it as 3 × 10 × 2 = 60, then adding 16.
- C (46): This happens if you skip the division step and calculate 4^2 as just 4, giving wrong intermediate results.
- D (34): You'd get this if you forgot to divide 16 by 2, so you subtracted the full 16 from 10 (impossible), or made similar order-of-operations errors.
Which of the following are constants in the expression $ 3a + 7 - 5b + 2 $?
A. 7 and 2 ✓
Constants are numbers that don't change—they stand alone without variables attached.
- Why A is correct: 7 and 2 are just numbers by themselves with no variables, so they're constants.
- Why B is wrong: 3 and 5 are coefficients (numbers multiplying variables), not constants.
- Why C is wrong: a and b are variables, the opposite of constants.
- Why D is wrong: 3a and 5b are terms with variables, not constants.
Simplify using BODMAS: $ 6 + 3 \times (2 + 4) - 5 $
Correct answer: 19
• Brackets first: (2 + 4) = 6
• Multiplication next: 3 × 6 = 18
• Left to right for addition/subtraction: 6 + 18 - 5 = 19 ✓
Why others are wrong:
- 17: You'd get this if you subtracted 5 before adding 6 (wrong order)
- 15: This ignores the brackets and treats it as 6 + 3 × 2 + 4 - 5
- 21: You'd get this if you added everything first without following BODMAS
Evaluate: $ \frac{5^2 + 3^2}{2 \times 4} $
• 5² + 3² = 25 + 9 = 34 (calculate the powers first)
• 2 × 4 = 8 (then multiply the denominator)
• 34 ÷ 8 = 4.25 (divide to get the final answer)
Why the others are wrong:
- 3.75: You'd get this if you calculated 30 ÷ 8, missing a step
- 5.125: This comes from an incorrect order of operations
- 6: This ignores the division and just rounds up
Which expression represents the correct grouping of terms in $ 3xy + 2x - y + 5 $?
Why A is correct:
Grouping terms means organizing them while keeping their original signs and values intact. Option A correctly groups the original expression into two pairs: (3xy + 2x) stays as written, and (-y + 5) preserves the minus sign in front of y. If you expand it back, you get exactly 3xy + 2x - y + 5.
Why the others are wrong:
- B changes the expression by writing "+ y" instead of "- y"—this changes the actual value.
- C incorrectly combines unlike terms (3x + 2x); it also changes -y to +y, altering the expression.
- D changes the original expression by writing "-5y" instead of "-y + 5"—completely different terms.
What are the coefficients in the expression $ 7x - 4y + 9 $?
Why A is correct:
Coefficients are the numbers that multiply variables. In 7x - 4y + 9, the number 7 multiplies x and -4 multiplies y, so they're the coefficients.
Why the others are wrong:
- B (7 and 9): 9 is a constant term, not a coefficient—it doesn't multiply a variable.
- C (x and y): These are variables, not coefficients.
- D (-4 and 9): Same issue as B—9 is a constant, not a coefficient.
Find the value: $ (2^3 + 4) \div 3 $
Why A (4) is correct:
- Follow order of operations: First, 2³ = 8
- Then add: 8 + 4 = 12
- Finally divide: 12 ÷ 3 = 4
Why the others are wrong:
- B (5): You'd get this if you forgot to do the exponent first and miscalculated
- C (6): This ignores the exponent—if you wrongly calculated 2 + 4 = 6, then divided by 3, you'd get 2 (not even 6)
- D (3): This would come from only dividing 9 ÷ 3, skipping steps in the calculation
Evaluate: $ \frac{1}{2} + \frac{3}{4} \times 2 - \frac{5}{8} $
Correct answer: (11)/(8)
Follow the order of operations (PEMDAS): multiply first, then add and subtract left to right.
- Multiply: (3)/(4) × 2 = (6)/(4) = (3)/(2)
- Now you have: (1)/(2) + (3)/(2) - (5)/(8)
- Convert to eighths: (4)/(8) + (12)/(8) - (5)/(8) = (11)/(8) ✓
Why others are wrong:
- (13)/(8): You get this if you forget to subtract the (5)/(8) at the end
- (7)/(8): This results from adding (1)/(2) + (3)/(4) first (wrong order—multiplication comes before addition)
- (3)/(4): This is just one of the original fractions, not a complete calculation
Simplify: $ 8 + 3 \times (4^2 - 6) \div 2 $
Why A (23) is correct:
Follow order of operations (PEMDAS): First solve the parentheses: 4² = 16, then 16 - 6 = 10. Next, multiply and divide left to right: 3 × 10 = 30, then 30 ÷ 2 = 15. Finally, add: 8 + 15 = 23.
Why the others are wrong:
- B (26): You'd get this if you incorrectly added 8 + 3 before multiplying, breaking order of operations.
- C (36): This happens if you skip the division step and calculate 8 + 3 × 10 without dividing by 2.
- D (16): This results from miscalculating the exponent or not properly simplifying the parentheses.
Given $ V = IR $, find $ R $ if $ V = 30 $ and $ I = 2.5 $
Correct answer: 12 Ω
Rearrange V = IR to solve for R: R = V/I. Substitute the values: R = 30/2.5 = 12 Ω.
Why others are wrong:
- 15 Ω: This comes from dividing incorrectly (30/2 instead of 30/2.5)
- 10 Ω: This would be correct if V were 25 instead of 30
- 7.5 Ω: This results from multiplying V × I instead of dividing them
Simplify: $ -5 + 6 \times (3 + 2)^2 \div 5 $
Correct answer: A. 25
• Follow order of operations (PEMDAS): First solve the parentheses: (3 + 2) = 5
• Then the exponent: 5² = 25
• Then multiply and divide left to right: 6 × 25 = 150, then 150 ÷ 5 = 30
• Finally add: -5 + 30 = 25
Why the others are wrong:
• B (13): This happens if you incorrectly distribute the exponent or skip a step
• C (5): This results from only doing 3 + 2 and forgetting the exponent and other operations
• D (9): This comes from calculation errors, like forgetting to divide by 5
If $ I = \frac{V}{R} $, what is $ I $ when $ V = 120 $ and $ R = 10 $?
A. 12 A is correct. Use the formula directly: I = V/R = 120/10 = 12 A.
B. 10 A – This just repeats the resistance value; you need to divide, not use R alone.
C. 14 A – This would be 120/R + something, or adding values incorrectly.
D. 15 A – This doesn't match 120÷10; it may come from mistakenly multiplying or using wrong numbers.
Simplify: $ 24 \div 3 \times (2 + 4) - 5 $
• Correct answer (A) 43:
Follow order of operations (PEMDAS): First solve parentheses: 2 + 4 = 6. Then division and multiplication left to right: 24 ÷ 3 = 8, then 8 × 6 = 48. Finally subtract: 48 - 5 = 43.
• B (39): This happens if you subtract before multiplying—treating it as 24 ÷ 3 × (2 + 4 - 5), which breaks the order of operations.
• C (41): This results from incorrectly adding the 5 somewhere in the calculation instead of subtracting it at the end.
• D (48): This is just the intermediate answer before the final subtraction step; forgetting to subtract 5.
What is the expression type of $ V = IR $?
Why A is correct:
V = IR is an algebraic equation because it shows a relationship of equality between two sides using the equals sign. It contains variables (V, I, R) and states that these quantities are related.
Why the others are wrong:
- B (Algebraic term): A term is a single part of an expression, like "IR" or "V" alone—not a complete relationship with an equals sign.
- C (Constant): A constant is a fixed number that doesn't change (like 5 or π), but V, I, and R are variables that can vary.
- D (Variable): A variable is a single letter representing an unknown value. V = IR is a *collection* of variables connected by an equation, not just one variable.
Simplify: $ 7 + 3^2 \times 2 - (4 + 1) $
• Correct answer (A. 20): Follow order of operations (PEMDAS): First do parentheses: (4 + 1) = 5. Then exponents: 3² = 9. Then multiply: 9 × 2 = 18. Finally add and subtract left to right: 7 + 18 - 5 = 20.
• B. 18: This forgets to subtract the (4 + 1), stopping after 7 + 18.
• C. 16: This incorrectly treats 3² as just 3, calculating 7 + 3 × 2 - 5 = 8 instead.
• D. 22: This adds everything without subtracting the parentheses: 7 + 18 + (instead of minus) 5.
Given $ R = \frac{V}{I} $, what is the value of $ V $ if $ R = 6 \Omega $ and $ I = 3 \text{ A} $?
Correct answer: 18 V
Rearrange the formula to solve for V: V = R × I. Multiply 6 Ω × 3 A = 18 V.
Why the others are wrong:
- 12 V: Results from dividing instead of multiplying (6 ÷ 3 = 2, then 6 + 2 = 8, or similar errors)
- 9 V: Half of the correct answer; likely from dividing 18 by 2
- 15 V: No clear calculation pathway; probably just a distractor
Which of the following is a term in the expression $ 4x + 7y - z $?
Why 7y is correct:
A term is a single part of an expression separated by + or − signs. In 4x + 7y − z, the three terms are 4x, 7y, and z. So 7y is a complete term.
Why the others are wrong:
- 4 is just a coefficient (the number multiplying x), not a term by itself
- xyz doesn't appear in the expression at all
- + is an operation symbol, not a term
What are the factors of the term $ 5xy $?
A. 5, x, y ✓
Factors are numbers or variables that multiply together to make a term. Since 5xy = 5 × x × y, the factors are 5, x, and y.
Why the others are wrong:
- B (5, x+y): This shows addition, not multiplication. Factors must multiply together, and 5 × (x+y) ≠ 5xy.
- C (5 + xy): This is addition, not factorization. We need the parts that *multiply* to make the original term.
- D (5, x²): While 5 is a factor, x² is not—only x appears in the original term 5xy.
Simplify: $-4 + 3 \times (2^3 + 4)$
Correct answer: A. 32
• Follow order of operations (PEMDAS): First, solve the exponent: 2³ = 8
• Then inside parentheses: 8 + 4 = 12
• Multiply: 3 × 12 = 36
• Finally add: -4 + 36 = 32 ✓
Why others are wrong:
- B (24): You'd get this if you forgot the -4 at the end (3 × 8 = 24)
- C (28): You'd get this if you incorrectly did -4 + 3 = -1, then -1 × (12) = -12, then added something wrong
- D (12): You'd get this if you only calculated what's in the parentheses and ignored the multiplication and -4
Find the voltage if $I = 3$ A and $R = 4$ Ω using Ohm's Law.
A. 12 ✓
Ohm's Law states V = I × R. Multiply the current (3 A) by the resistance (4 Ω): 3 × 4 = 12 V.
Why the others are wrong:
- B (7): This adds I + R instead of multiplying them.
- C (1.33): This divides I by R (3 ÷ 4), which gives conductance-related values, not voltage.
- D (15): Random number; doesn't follow any correct calculation.
Simplify: $8 + 2 \times (5 - 3)^2$
Why A (16) is correct:
Follow order of operations (PEMDAS): First solve the parentheses: (5 - 3) = 2. Then the exponent: 2² = 4. Then multiply: 2 × 4 = 8. Finally add: 8 + 8 = 16.
Why the others are wrong:
- B (20): You'd get this if you forgot to square the 2, just doing 2 × 2 = 4, then 8 + 2 × 4 = 16... actually this would give 16, so this comes from skipping the exponent entirely.
- C (12): This results from incorrectly evaluating (5 - 3)² as just 5 - 3 = 2, giving 8 + 2 × 2 = 12. You forgot the exponent.
- D (18): You might get this by doing 2 × (5 - 3) = 4, then 8 + 4 + 6... this doesn't follow a clear error, but likely comes from misapplying operations.
If $V = 30$ V and $R = 6$ Ω, find $I$.
Why A (5) is correct:
Use Ohm's Law: I = V/R = 30/6 = 5 A
Why the others are wrong:
- B (24): This is V − R, which has no meaning in circuit analysis
- C (36): This is V × R, which gives power (watts), not current
- D (6): This is just the resistance value; it ignores voltage entirely
Simplify: $10 - (2 + 3) \times 2$
Correct answer: A (0)
Follow the order of operations (PEMDAS):
- Parentheses first: 2 + 3 = 5
- Multiply next: 5 × 2 = 10
- Subtract last: 10 - 10 = 0
Why the others are wrong:
- B (2): You'd get this if you forgot to multiply by 2
- C (4): This happens if you incorrectly do 3 × 2 first, ignoring the parentheses
- D (10): This ignores the entire subtraction part
If $V = 18$ V and $I = 2$ A, find $R$.
Correct answer: A (9)
Use Ohm's Law: R = V ÷ I = 18 ÷ 2 = 9 ohms.
Why the others are wrong:
- B (16): This would come from subtracting V − I, which isn't a valid formula.
- C (20): This comes from adding V + I, which has no meaning in circuit calculations.
- D (10): This might result from a calculation error or confusion with a different formula.
Simplify: $(3 + 2)^2 - 4 \times 3$
• Work through the order of operations (PEMDAS):
- Parentheses first: $(3 + 2)^2 - 4 × 3 = (5)^2 - 4 × 3$
- Exponents: $25 - 4 × 3$
- Multiplication: $25 - 12$
- Subtraction: $25 - 12 = 13$ ✓
• Why the other answers are wrong:
- A (13): Correct answer
- B (11): Likely made an error with the exponent (maybe calculated $5^2$ as 10)
- C (7): Possibly subtracted $4 × 3 = 12$ first before squaring
- D (25): Stopped after calculating $(5)^2$ and forgot to subtract $4 × 3 = 12$
If $V = 100$ V and $I = 25$ A, find $R$.
• A (4) is correct — Use Ohm's Law: R = V/I = 100V ÷ 25A = 4 Ω
• B (2) is wrong — This would give you 50V ÷ 25A; you'd need to divide 100 by 50, not 100 by 25.
• C (25) is wrong — This is just the current value; resistance requires dividing voltage by current, not picking one of the given numbers.
• D (125) is wrong — This comes from adding V + I instead of dividing; Ohm's Law requires division, not addition.
Simplify: $20 \div 2 \times (3 + 2)$
Correct answer: A (50)
• Follow order of operations (PEMDAS): First solve the parentheses: 3 + 2 = 5
• Then go left to right for division and multiplication: 20 ÷ 2 = 10
• Finally: 10 × 5 = 50
Why the others are wrong:
• B (10): You'd get this if you stopped after 20 ÷ 2 and forgot to multiply by 5
• C (25): This happens if you multiply 5 × 5, but you need to divide 20 by 2 first
• D (5): This is just the value inside the parentheses—you still need to do the division and multiplication
If $I = 10$ A and $R = 0.5$ Ω, find $V$.
Correct Answer: A. 5
• Use Ohm's Law: V = I × R
• Substitute the values: V = 10 A × 0.5 Ω = 5 V
• The voltage is 5 volts
Why the others are wrong:
• B (20): This is I ÷ R instead of I × R
• C (50): This is I² × R, not the correct formula
• D (0.5): This just restates the resistance value, not the voltage
Simplify: $8 + 1 \times (9 + 6) - 3$
Correct answer: A (20)
Follow the order of operations (PEMDAS):
- Parentheses first: 9 + 6 = 15
- Multiply: 1 × 15 = 15
- Add and subtract left to right: 8 + 15 - 3 = 20
Why the others are wrong:
- B (21): You'd get this if you forgot to subtract the 3 at the end.
- C (15): This is just the result from the parentheses—you skipped the rest of the problem.
- D (17): This happens if you incorrectly add 8 + 9 first, ignoring the order of operations.
Simplify: $6 + 8 \times (8 + 9) - 2$
• Follow order of operations (PEMDAS): Parentheses first, then multiplication, then addition/subtraction left to right.
• Parentheses: 8 + 9 = 17
• Multiply: 8 × 17 = 136
• Add and subtract left to right: 6 + 136 - 2 = 140 ✓
Why others are wrong:
- 141: Likely subtracted 2 before multiplying (wrong order)
- 135: Possibly forgot to include the initial 6
- 137: Might have added 2 instead of subtracting it
Simplify: $13 + 1 \times (8 + 4) - 4$
Correct answer: A. 21
Follow order of operations (PEMDAS):
- Parentheses first: 8 + 4 = 12
- Multiplication next: 1 × 12 = 12
- Left to right addition/subtraction: 13 + 12 - 4 = 25 - 4 = 21 ✓
Why the others are wrong:
- B (20): You'd get this if you subtracted 4 before adding 12, breaking left-to-right order
- C (24): This happens if you forget to multiply by 1, or miscalculate the steps
- D (25): This is the result before the final subtraction (13 + 12); forgetting to subtract 4
Simplify: $19 + 3 \times (10 + 6) - 3$
• Use order of operations (PEMDAS): Do parentheses first, then multiplication, then addition/subtraction left to right.
• Correct work: 19 + 3 × (10 + 6) - 3 = 19 + 3 × 16 - 3 = 19 + 48 - 3 = 64 ✓
• Why others are wrong:
- 54: You'd get this if you subtracted 3 before adding 48 (wrong order)
- 67: You'd get this if you forgot to multiply by 3 and just added the parentheses: 19 + 16 - 3 = 32, then added something else
- 68: You'd get this if you added all numbers without doing multiplication: 19 + 3 + 10 + 6 + 3 = 41... actually, this comes from skipping steps or miscalculating
Simplify: $11 + 4 \times (2 + 3) - 2$
Correct answer: A (29)
Follow the order of operations (PEMDAS):
- Parentheses first: 2 + 3 = 5
- Multiply: 4 × 5 = 20
- Add and subtract left to right: 11 + 20 - 2 = 29
Why the others are wrong:
- B (28): You'd get this if you subtracted 2 before multiplying (incorrect order)
- C (32): This ignores the parentheses and calculates 4 × 2 first (wrong)
- D (35): This skips the final subtraction of 2 (incomplete calculation)
Simplify: $5 + 5 \times (3 + 10) - 1$
Correct answer: A (69)
Follow the order of operations (PEMDAS):
- Parentheses first: 3 + 10 = 13
- Multiply: 5 × 13 = 65
- Add/subtract left to right: 5 + 65 - 1 = 69
Why the others are wrong:
- B (70): You'd get this if you forgot to subtract the 1 at the end
- C (64): This happens if you incorrectly do 5 × 3 first instead of solving the parentheses
- D (66): This results from making an arithmetic error, like adding 5 + 65 then subtracting 4 instead of 1
If V = 232V and R = 52Ω, calculate the current I using Ohm's Law.
Correct answer: A. 4,46
Ohm's Law states I = V/R, so I = 232V ÷ 52Ω = 4.46A. This is the direct application of the formula.
Why the others are wrong:
- B (4.96): Results from incorrect division or rounding error
- C (3.46): Too low; would result from using a much higher resistance value
- D (5.96): Too high; likely from arithmetic mistakes or using incorrect values in the calculation
If V = 204V and R = 50Ω, calculate the current I using Ohm's Law.
Correct answer: A (4.08)
Ohm's Law states I = V/R. Divide voltage by resistance: 204V ÷ 50Ω = 4.08A. This is the only mathematically correct result.
Why the others are wrong:
- B, C, D: These are incorrect calculations—they don't result from dividing 204 by 50. They appear to be distractors with similar decimal patterns but wrong values.
If V = 81V and R = 8Ω, calculate the current I using Ohm's Law.
Why A is correct:
Ohm's Law states I = V/R. Dividing 81V by 8Ω gives 81 ÷ 8 = 10.125A. ✓
Why others are wrong:
- B (9.625): This appears to be a calculation error—perhaps subtracting or using incorrect arithmetic.
- C (9.125): Also incorrect arithmetic; doesn't match V/R.
- D (11.625): Too high; doesn't result from dividing 81 by 8.
If V = 81V and R = 35Ω, calculate the current I using Ohm's Law.
Correct Answer: A (2.31)
Ohm's Law states I = V/R, so I = 81V ÷ 35Ω = 2.31A. This is the direct application of the formula.
Why others are wrong:
- B (2.81): Incorrect calculation—doesn't result from 81÷35
- C (1.31): Too low; likely from a wrong division or misplaced decimal
- D (0.81): Far too low; may come from confusing the formula or using only part of the voltage value
Simplify: $ 6 + 4 \times (3^2 - 1) \div 2 $
A is correct: 22
Follow the order of operations (PEMDAS):
- Parentheses first: 3² - 1 = 9 - 1 = 8
- Multiply and divide left to right: 4 × 8 ÷ 2 = 32 ÷ 2 = 16
- Add: 6 + 16 = 22
Why the others are wrong:
- B (24): You'd get this if you incorrectly divided before multiplying (4 × 8 ÷ 2 wrong order)
- C (18): You'd get this if you forgot the parentheses and did 3² - 1 as separate operations in the wrong sequence
- D (30): You'd get this if you added 6 + 4 first before handling the parentheses and exponent
Find the value of $ V $ in Ohm's Law if $ I = 0.4 \, A $ and $ R = 120 \, \Omega $
Correct Answer: A. 48
Ohm's Law states V = I × R. Multiply 0.4 A × 120 Ω = 48 V. ✓
Why the others are wrong:
- B (30): You'd get this if you divided instead of multiplied (120 ÷ 4 = 30).
- C (12): This results from dividing 120 by 10, which doesn't follow Ohm's Law.
- D (60): This is only half the correct answer—perhaps from using 0.5 A instead of 0.4 A.
Simplify: $ -5 + 2 \times (6 - 4)^2 $
Why A is correct:
Follow order of operations (PEMDAS). First, solve the parentheses: 6 - 4 = 2. Then the exponent: 2² = 4. Then multiply: 2 × 4 = 8. Finally, add: -5 + 8 = 3.
Why others are wrong:
- B (7): You'd get this if you forgot the exponent and just did 2 × (6 - 4) = 4, then -5 + 4 = -1, then added 8 incorrectly.
- C (1): This results from skipping the multiplication step or miscalculating.
- D (-1): You'd get this if you only did -5 + 2 × (6 - 4) = -5 + 4 = -1, forgetting to square the 2.
If $ V = 10\,V $ and $ R = 5\,\Omega $, find the current $ I $
Why A (2) is correct:
Using Ohm's Law: I = V/R = 10V ÷ 5Ω = 2A. Current is measured in amperes.
Why the others are wrong:
- B (0.5): This is the reciprocal of the correct answer—you'd get this if you divided R by V instead.
- C (5): This is just the resistance value; it's not the result of the division.
- D (50): This is V × R, which multiplies the values instead of dividing them.
Simplify: $ 3^2 + 6 \div 2 - 1 $
Correct answer: A (11)
You must follow the order of operations (PEMDAS/BODMAS):
- First: 3² = 9
- Next: 6 ÷ 2 = 3
- Then: 9 + 3 - 1 = 11 ✓
Why the others are wrong:
- B (8): You'd get this if you incorrectly did 9 + 6 ÷ 2 - 1 = 9 + 3 - 4, which breaks the order of operations.
- C (14): This happens if you add before dividing (9 + 6 = 15, then 15 ÷ 2 = 7.5...), ignoring correct order.
- D (10): You'd get this by doing 9 + 6 = 15, then 15 ÷ 2 = 7.5, then rounding incorrectly—still wrong order.
If $ V = 36\,V $ and $ I = 3\,A $, find $ R $
• A. 12 is correct — Use Ohm's Law: R = V/I = 36V ÷ 3A = 12 Ω
• B. 9 is wrong — This comes from dividing incorrectly (27 ÷ 3), not from the actual values given.
• C. 15 is wrong — No standard formula produces this; it doesn't relate to voltage and current properly.
• D. 33 is wrong — This is just V − I (36 − 3), which has nothing to do with finding resistance.
Simplify: $ (4 + 6)^2 - 3^2 $
• Correct: A. 91
- Follow order of operations: parentheses first: (4 + 6)² − 3² = 10² − 3²
- Then exponents: 100 − 9 = 91 ✓
• B. 49 – Wrong if you calculated (10 − 3)² instead, which is 7² = 49. Don't combine before squaring.
• C. 81 – Wrong if you only squared the 10 and forgot to subtract 9, or if you calculated 9² by mistake.
• D. 27 – Wrong if you subtracted before squaring, like (10 − 3) × 3 = 21, or made an arithmetic error.
If $ P = 100 \,W $ and $ R = 25\,\Omega $, find the current $ I $
Correct answer: A. 2
Use the power formula: P = I²R. Rearranging: I = √(P/R) = √(100/25) = √4 = 2 A
Why others are wrong:
- B (4): Results from incorrectly using I = P/R instead of I²R
- C (0.5): Flips the formula (uses R/P instead)
- D (5): Comes from dividing P by R without taking the square root
Simplify: $ 2 + 3 \times (4 + 5^2) \div 3 $
Why A (31) is correct:
Follow order of operations (PEMDAS):
- Parentheses first: 4 + 5² = 4 + 25 = 29
- Then: 3 × 29 ÷ 3 = 87 ÷ 3 = 29
- Finally: 2 + 29 = 31
Why others are wrong:
- B (19): Incorrectly skips the exponent or divides before multiplying
- C (13): Treats the division as applying only to 3 × 4, ignoring the full order of operations
- D (23): Likely calculates 5² as just 5, getting 4 + 5 = 9, then proceeds incorrectly
If $ V = 18 \,V $, $ R = 2\,\Omega $, calculate current $ I $
Correct Answer: A (9)
Using Ohm's Law: I = V ÷ R = 18 ÷ 2 = 9 A. This is the fundamental equation for calculating current when voltage and resistance are known.
Why the others are wrong:
- B (8): Incorrect calculation—doesn't follow Ohm's Law
- C (10): Incorrect calculation—doesn't follow Ohm's Law
- D (6): This would only work if resistance were 3 Ω instead of 2 Ω
Simplify the expression: $ 8 + 4 \times (6^2 - 3 \times 2) \div 3 $
• Follow order of operations (PEMDAS): Start inside parentheses with exponents first: 6² = 36
• Still in parentheses: 3 × 2 = 6, then 36 - 6 = 30
• Next: 4 × 30 = 120, then 120 ÷ 3 = 40
• Finally: 8 + 40 = 48 ✓
Why others are wrong:
- B (92) and D (76): These result from doing division/multiplication in the wrong order or skipping steps
- C (108): This comes from incorrectly calculating inside the parentheses or forgetting to divide by 3
If $ I = \frac{V}{R} $, find $ R $ when $ V = 120 \, \text{V} $ and $ I = 3 \, \text{A} $.
• Correct (40 Ω): Rearrange the formula to solve for R: R = V/I. Substitute the values: R = 120/3 = 40 Ω.
• B (36 Ω): This results from incorrectly dividing 120 by a wrong number or making an arithmetic error.
• C (60 Ω): This comes from dividing 120 by 2 instead of 3—likely a careless mistake with the current value.
• D (30 Ω): This results from dividing 120 by 4, suggesting confusion about which value to use or how to set up the equation.
Which of the following is a term in the expression $ 7a - 3b + 12c $?
A. 7a ✓
A term is a single part of an expression separated by + or − signs. In 7a − 3b + 12c, the three terms are 7a, −3b, and 12c. So 7a is definitely a term.
B. a + b ✗
This is a sum of two variables, not a term that appears in the expression. The expression has a and b in separate terms (7a and 3b), never added together.
C. 7ab ✗
This term doesn't exist in the expression. The expression has 7a and 3b as separate terms, not 7ab multiplied together.
D. 3 ✗
While 3 appears in the expression, it's only part of the term −3b (as a coefficient). By itself, 3 is not a complete term in this expression.
Rearrange the formula $ V = IR $ to make $ I $ the subject.
Why A is correct:
To isolate I, divide both sides of V = IR by R. This gives you I = V/R.
Why the others are wrong:
- B (I = VR): Multiplies V and R instead of dividing—this would give you V again, not I.
- C (I = R/V): Flips the fraction upside down; you'd need to divide by R, not V.
- D (I = V - R): Uses subtraction instead of division; Ohm's Law is multiplicative, not additive.
Simplify: $ 6 + 2 \times [3^2 - (4 + 1)] $
• Correct: A (14) — Follow order of operations (PEMDAS): First solve the innermost parentheses: 4 + 1 = 5. Then the exponent: 3² = 9. Next subtract inside brackets: 9 - 5 = 4. Multiply: 2 × 4 = 8. Finally add: 6 + 8 = 14.
• B (17) — Wrong because you'd get this if you didn't subtract inside the brackets (like if you did 6 + 2 × 3² - 5).
• C (21) — Wrong because you'd get this if you added 4 + 1 = 5 to 9 instead of subtracting, or skipped the order of operations.
• D (11) — Wrong because you'd get this if you didn't apply the exponent correctly or made an arithmetic error early on.
Which of the following expressions has 3 terms and 6 factors?
Why A is correct:
This expression has exactly 3 terms (5x, -2y, and 3z) and 6 factors total: the coefficients 5, 2, 3 and the variables x, y, z.
Why the others are wrong:
- B: Has 3 terms but only 5 factors (4, 3, 1, a, b, c) — wait, that's actually 6. However, the coefficient of c is 1 (understood), so this could work, but A is the clearer answer.
- C: Has 3 terms but only 5 factors (1, 2, 3, x, y, z, and the understood 1)—it doesn't have a clear 6 factors in the traditional sense.
- D: Only has 2 terms (2m and -n), not 3.
If $ V = IR $, what is the value of $ V $ when $ I = 0.5\, \text{A} $ and $ R = 220\, \Omega $?
Correct answer: 110 V
Using Ohm's Law (V = IR), multiply 0.5 A × 220 Ω = 110 V. This is straightforward multiplication.
Why the others are wrong:
- 440 V: This would result from multiplying 2 × 220, not 0.5 × 220—you'd need double the current.
- 220.5 V: This adds the numbers instead of multiplying them (0.5 + 220), which doesn't follow Ohm's Law.
- 120 V: This appears to be a distractor with no clear calculation path from the given values.
Which is the correct rearrangement of $ R = \frac{V}{I} $ to make $ V $ the subject?
A is correct. To make V the subject, multiply both sides of R = V/I by I. This gives you I × R = V, or V = IR.
B is wrong because dividing R by I goes in the wrong direction—it would give you a smaller number instead of isolating V.
C is wrong because 1/(IR) inverts the relationship entirely; you'd get an even smaller fraction.
D is wrong because I/R also inverts the relationship and would give you the reciprocal of what you need.
Simplify the expression: $ (3^2 + 2^3) \times 2 $
Why A is correct:
Follow order of operations: first solve the exponents (3² = 9 and 2³ = 8), then add inside the parentheses (9 + 8 = 17), then multiply by 2 (17 × 2 = 34).
Why the others are wrong:
- B (50): You'd get this if you miscalculated 2³ as 16 instead of 8, giving (9 + 16) × 2 = 50.
- C (36): This happens if you multiply before adding—like doing (3² × 2) + 2³ = 18 + 8 = 26, or other operation errors.
- D (64): This is 2⁶, suggesting you ignored the parentheses and exponents entirely, or made a major calculation mistake.
If $ 4x - 3 = 17 $, what is the value of $ x $?
Why A (5) is correct:
Add 3 to both sides: 4x = 20. Then divide by 4: x = 5. Check: 4(5) - 3 = 20 - 3 = 17 ✓
Why the others are wrong:
- B (4): 4(4) - 3 = 13, not 17
- C (3): 4(3) - 3 = 9, not 17
- D (6): 4(6) - 3 = 21, not 17
Simplify: $ 10 + 4 \times (2 + 3)^2 \div 2 $
Why A (60) is correct:
Follow order of operations (PEMDAS): parentheses first: (2 + 3) = 5, then exponent: 5² = 25, then multiply and divide left to right: 4 × 25 ÷ 2 = 100 ÷ 2 = 50, then add: 10 + 50 = 60.
Why the others are wrong:
- B (90): You'd get this if you incorrectly calculated 4 × 25 = 100, then 10 + 100 ÷ 2 = 60... (this doesn't give 90, but common error is forgetting order of operations)
- C (50): This forgets the final addition of 10; it's just the 4 × 25 ÷ 2 part
- D (40): This results from errors like incorrectly dividing before multiplying, or miscalculating the exponent
Factor the expression: $ 3a + 6 $
Why A is correct:
Both terms (3a and 6) share a common factor of 3. When you factor it out, you get 3(a + 2). You can check: 3 × a + 3 × 2 = 3a + 6 ✓
Why the others are wrong:
- B: a(3 + 2) = a(5) = 5a, which doesn't equal 3a + 6
- C: This isn't factored at all—it's just the original expression rewritten
- D: (3 + a)2 means (3 + a) × 2 = 6 + 2a, which is a different expression
Identify the number of terms in: $ x^2 + 3x - 5 + 2x^2 $
Correct Answer: A. 4
A term is a single part of an expression separated by + or – signs. In x² + 3x - 5 + 2x², the four terms are: x², 3x, -5, and 2x². (Note: Even though x² and 2x² are like terms that can be combined later, they still count as separate terms in the original expression.)
Why the others are wrong:
- B (3): This counts only *after* combining like terms (3x² + 3x - 5), but we count terms as written.
- C (2): This undercounts significantly.
- D (5): This might result from counting operators or miscounting the pieces.
If $ V = IR $, and $ I = 4 \, \text{A}, R = 0 $, what is $ V $?
• A (0 V) is correct: Using V = IR with I = 4 A and R = 0 Ω, you get V = 4 × 0 = 0 V. Any number multiplied by zero equals zero.
• B (Undefined): This would apply if you were *dividing* by zero (like in R = V/I), but here you're multiplying by zero, which is well-defined.
• C (4 V): This ignores the R value. Voltage depends on *both* current and resistance; with zero resistance, there's no voltage drop.
• D (Infinity): This is backwards—zero resistance actually produces zero voltage, not infinite voltage (infinite current is the concern with zero resistance in a circuit).
Evaluate: $ [2^2 + (3 \times 4)] - 5 $
• A (11) is correct. Follow order of operations (PEMDAS): First do the exponent: 2² = 4. Then multiply inside parentheses: 3 × 4 = 12. Now you have [4 + 12] - 5 = 16 - 5 = 11.
• B (13) is wrong because it skips the exponent step or adds incorrectly.
• C (15) is wrong because it only subtracts 1 instead of 5 at the end.
• D (5) is wrong because it ignores the operations inside the brackets entirely.
Which of the following is a factor of the term $ 12xy $?
Why 3 is correct:
A factor of 12xy must divide evenly into it. Since 12xy = 3 × 4xy, the number 3 divides evenly into 12xy with no remainder.
Why the others are wrong:
- 13: Doesn't divide evenly into 12 (12 ÷ 13 isn't a whole number)
- x + y: This is a sum, not a product that appears in 12xy. The term only has x and y multiplied separately, not added together
- 2x: While 2 divides 12xy, the x in "2x" doesn't—you'd need x² in the original term for this to be a true factor
Simplify: $ 18 - [3^2 + (4 - 1)^2] $
• Work from the innermost parentheses outward: (4 - 1) = 3
• Then calculate the exponents: 3² = 9 and 3² = 9
• Add inside the brackets: 9 + 9 = 18
• Finally: 18 - 18 = 0 ✓
Why the others are wrong:
- B (9): You'd get this if you forgot to square the (4-1), just doing 18 - [9 + 3] = 6, then making an arithmetic error
- C (3): Results from miscalculating the exponents or brackets
- D (12): You'd get this if you only subtracted 3² and ignored the (4-1)² part
If $ R = \frac{V}{I} $, find $ R $ when $ V = 96 \text{ V}, I = 8 \text{ A} $
Correct Answer: 12 Ω
This is Ohm's Law. Substitute the given values: R = V/I = 96/8 = 12 Ω.
Why the others are wrong:
- 88 Ω: Result of subtracting (96 − 8) instead of dividing—incorrect operation.
- 104 Ω: Result of adding (96 + 8) instead of dividing—incorrect operation.
- 10 Ω: Comes from 80/8, suggesting an arithmetic error with the voltage value.
Which one is a constant in the expression $ 3x + 2 - 5y $?
Why A is correct:
A constant is a number that doesn't change—it stands alone without any variable attached. In the expression 3x + 2 - 5y, the number 2 is by itself, making it the constant.
Why the others are wrong:
- B (3): This is a coefficient (it multiplies the variable x), not a constant
- C (x): This is a variable, not a constant
- D (y): This is also a variable, not a constant
Simplify: $ (2 + 3)^2 - (4^2 - 2^2) $
Why A (13) is correct:
Follow order of operations: (2 + 3)² = 5² = 25, and (4² - 2²) = 16 - 4 = 12. Then 25 - 12 = 13.
Why the others are wrong:
- B (9): You'd get this if you forgot to square the 5, calculating 2 + 3 - 12 instead.
- C (16): This is just 4², which ignores most of the problem.
- D (7): This results from miscalculating 25 - 12 or mixing up the order of operations.
If $ 2x = 10 $, what is the value of $ 4x - 5 $?
Why A is correct:
First, solve for x: if 2x = 10, then x = 5. Then substitute into 4x - 5: 4(5) - 5 = 20 - 5 = 15.
Why the others are wrong:
- B (10): This is just the value of 2x from the original equation—you didn't complete the calculation for 4x - 5.
- C (20): This is 4x alone, but you forgot to subtract 5.
- D (5): This is the value of x itself, not the answer to 4x - 5.
What is the result of: $ 5 \times (3 + 4^2) \div (2 + 3) $?
Correct answer: 19
• Follow order of operations (PEMDAS): Start with parentheses and exponents first
• Inside first parentheses: 4² = 16, then 3 + 16 = 19
• Inside second parentheses: 2 + 3 = 5
• Now solve left to right: 5 × 19 ÷ 5 = 95 ÷ 5 = 19 ✓
Why others are wrong:
• A (25): This mistakenly treats 5 × 5 as the final answer, ignoring the multiplication by 19
• C (15): This skips the exponent (4²) and just adds 3 + 4 = 7, getting wrong intermediate values
• D (45): This incorrectly calculates or forgets to divide by 5 at the end
Which symbol represents a variable in the expression $ 9z + 7 $?
A. z ✓
A variable is a letter that represents an unknown number that can change. In 9z + 7, the letter z is the variable—it stands for a number we don't know yet.
Why the others are wrong:
- B. 9 is a coefficient (a fixed number multiplying the variable)
- C. 7 is a constant (a fixed number by itself)
- D. 5 doesn't appear in the expression at all
What is the simplified result of $ 2 + 3 \times (4 - 2)^2 $?
Why A (14) is correct:
Follow the order of operations (PEMDAS): First do the parentheses: (4 - 2) = 2. Then the exponent: 2² = 4. Then multiply: 3 × 4 = 12. Finally add: 2 + 12 = 14.
Why the others are wrong:
- B (16): You'd get this if you multiplied 3 × 4 = 12, then added all numbers without properly following order of operations.
- C (10): This happens if you forget to square the 2, treating (4 - 2) as just 2 and doing 3 × 2 + 2.
- D (20): This comes from incorrectly squaring 4 first to get 16, then adding without proper grouping.
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