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📝 Division Property of Equality (13 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 13 questions available

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What is Division Property of Equality?

Definition:
The Division Property of Equality states that if you divide both sides of an equation by the same nonzero number, the equality is maintained. Formally, if $a = b$ and $c \neq 0$ , then $a/c = b/c$ . It is specifically used to eliminate multiplication by a constant.

Working:
When you have an equation like $4x = 20$ , divide both sides by 4 to isolate $x$ : $4x/4 = 20/4$ , resulting in $x = 5$ . The division cancels out the coefficient 4.

Example:
Solve $9z = 63$ using Division Property. Divide both sides by 9: $9z/9 = 63/9$ , so $z = 7$ . Check: $9(7) = 63$ .

Reason:
This property is the inverse of multiplication, allowing us to undo multiplication and solve for the variable when it has a coefficient.

Easy

Medium

Hard

📝 All Division Property of Equality MCQs

Q1. If $-7x = 42$ , what is the value of $x$ based on the division property of equality?

x = -6

✅

x = 6

x = -7

x = 294

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The division property states that if you divide both sides of an equation by the same nonzero number, the equality remains true. Here, dividing both sides by $-7$ gives $x = -6$ . It is a direct application.

Q2. Which equation correctly represents the statement: 'A number multiplied by 4 gives 28. Find the number.'?

4x = 28

✅

x/4 = 28

4 + x = 28

4 - x = 28

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Translating a verbal statement to an algebraic equation requires understanding that 'multiplied by' indicates multiplication. $4x = 28$ is the correct model, and solving by division gives $x = 7$ . The other options change the operation.

Q3. A taxi charges a fixed fare of $\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 3: 5 \̲)̲ plus $\" style="color:#cc0000">5$ plus $\</span>2$ per mile. If the total fare is $\$25$ , which equation and solution correctly model the situation?

2m + 5 = 25

;

m = 10

✅

2m = 25

;

m = 12.5

5m + 2 = 25

;

m = 4.6

2m + 5 = 25

;

m = 15

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The fixed cost is added to the variable cost. So $2m + 5 = 25$ . Subtracting 5 gives $2m = 20$ , then dividing by 2 gives $m = 10$ . This applies the division property after a subtraction step.

Q4. A student solved $3x = 15$ as $x = 5$ . Another solved $-3x = 15$ as $x = -5$ . Which statement about their solutions is true?

A.Both are correct because they correctly divided by the coefficient. ✅

B.First is correct, second is incorrect because dividing by a negative changes sign.

C.Both are incorrect because division property doesn’t apply to negative numbers.

D.First is incorrect, second is correct because positive divided by positive.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The division property applies regardless of sign. For $3x = 15$ , dividing by 3 gives 5. For $-3x = 15$ , dividing by $-3$ gives $-5$ . Both students correctly applied the property, showing understanding that sign matters in the result.

Q5. If the graph of $y = 5x$ passes through the point where $y = 20$ , what is the x-coordinate? Interpret using division property.

x = 4

✅

x = 100

x = 15

x = 0.25

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: On the graph, $y = 20$ means $5x = 20$ . Dividing both sides by 5 gives $x = 4$ . This represents the input value that produces the output 20 on the line, linking graphical representation with algebraic solution.

Q6. Compare the two solution methods for $8x = 64$ : Method A divides by 8, Method B multiplies by $\frac{1}{8}$ . Which is correct?

A.Both are correct and equivalent ✅

B.Only Method A is correct

C.Only Method B is correct

D.Neither is correct

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Dividing by 8 and multiplying by its reciprocal $\frac{1}{8}$ are equivalent operations. Both yield $x = 8$ . This question tests the conceptual understanding that division is multiplication by the reciprocal.

Q7. What is the solution of $\frac{x}{-4} = 12$ ? Note: Division property is used differently here.

x = -48

✅

x = -3

x = 48

x = 3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Here $x$ is divided by $-4$ . To isolate $x$ , multiply both sides by $-4$ (which is the inverse operation, since division property can be extended). $x = -48$ . This tests if students recognize that division property can be applied inversely.

Q8. A physics student knows that Force = mass × acceleration, or $F = ma$ . If $F = 120$ N and $a = 15$ m/s², what is the mass?

m = 8

kg ✅

m = 1800

m = 105

m = 135

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Using $F = ma$ , substitute values: $120 = 15m$ . Dividing both sides by 15 gives $m = 8$ . This models real-world application of division property in a scientific formula.

Q9. A student writes $0x = 5$ and claims division property gives $x = 5$ . What is the error?

A.Division by zero is undefined, so no solution ✅

B.Division by zero gives zero, so

x = 0

C.Zero divided by zero is 1, so

x = 1

D.It is correct because any number divided by 0 is 0

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The division property requires dividing by a nonzero number. Since the coefficient is 0, dividing both sides by 0 is undefined. The equation $0x = 5$ has no solution. This identifies the critical misconception about division by zero.

Q10. Given the equation $4x + 8 = 24$ , a student subtracts 8 first to get $4x = 16$ , then divides by 4 to get $x = 4$ . What property is used in each step?

A.Subtraction property, then division property ✅

B.Division property, then subtraction property

C.Only division property used twice

D.Only subtraction property used

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: This question requires identifying two different properties. Subtracting 8 uses the subtraction property of equality; dividing by 4 uses the division property. It tests the sequence and conceptual understanding of multiple properties.

Q11. If $-2x = -18$ , which of the following is the result of dividing both sides by -2?

x = 9

✅

x = -9

x = 36

x = -36

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Dividing both sides by $-2$ gives $x = \frac{-18}{-2} = 9$ . The signs cancel, producing a positive result. This is a straightforward application of the division property.

Q12. Which of the following is NOT a valid use of the division property of equality?

A.Dividing both sides by 2 in

2x = 10

B.Dividing both sides by -3 in

-3x = 12

C.Dividing both sides by 0 in

0x = 0

✅

D.Dividing both sides by 0.5 in

0.5x = 7

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Dividing by zero is invalid in any equation, including $0x = 0$ , because division by zero is undefined. Although the equation is true for all $x$ , you cannot divide by 0 to solve it. This is a common error.

Q13. If $\frac{x}{3} = 7$ , solve for $x$ using the division property after multiplying both sides by 3. What is the result?

x = 21

✅

x = \frac{7}{3}

x = 3

x = 10

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Although the equation involves division, we multiply both sides by 3 to isolate $x$ . This is the inverse operation. The division property is used implicitly because multiplication and division are inverse operations. $x = 21$ .

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

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