📝 Divide Integers in algebraic expressions (14 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 14 questions available

←Previous Topic:Multiply Integers in algebraic expressions📝 Divide Integers in algebraic expressionsSimplify Expressions with Integers:Next Topic→

What is Divide Integers in algebraic expressions?

Definition:
Dividing integers in algebraic expressions involves splitting signed numbers into equal parts, where the quotient's sign follows the same rule as multiplication: positive if divisor and dividend have same sign, negative if different, and this applies to integer coefficients in fractions and rational expressions.

Working:
Divide the absolute values of the integers, then assign sign based on same/different signs; if the dividend is zero, quotient is zero unless divisor is zero (undefined); in expressions, divide coefficients and simplify fractions by canceling common factors.

Example:
Divide $(-36)$ by $9$ in expression $\frac{-36x}{9}$ .
Solution: $(-36) \div 9 = -4$ (different signs, negative), so $\frac{-36x}{9} = -4x$ .

Reason:
Integer division is vital for simplifying algebraic fractions, solving equations involving division, and understanding rates and ratios, ensuring proper handling of negative results in real-world applications like slope calculations.

Easy

Medium

Hard

📝 All Divide Integers in algebraic expressions MCQs

Q1. A temperature changes by $-48^\circ C$ over 6 equal time intervals. If the temperature changes by the same amount each interval, what is the change per interval?

-8^\circ C

✅

8^\circ C

-42^\circ C

42^\circ C

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The total change is $-48^\circ C$ and it is divided equally among 6 intervals. Thus, $-48\div6=-8$ . The negative sign indicates that the temperature decreases by $8^\circ C$ during each interval.

Q2. A student claims that $(-72)\div(-9)=-8$ because both numbers are negative. Which reasoning correctly evaluates the expression?

A.The quotient is

-8

because negative divided by negative stays negative.

B.The quotient is

8

because dividing two numbers with the same sign gives a positive result. ✅

C.The quotient is

-63

because the signs cancel through subtraction.

D.The quotient is

81

because both signs become positive before division.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When two integers have the same sign, their quotient is positive. Since $72\div9=8$ , the quotient of $-72$ and $-9$ is $8$ , not $-8$ . The student's error comes from applying the opposite-sign rule incorrectly.

Q3. A delivery company records a total adjustment of $-\$96$ across 8 identical transactions. If the adjustment is distributed equally, what does the result mean?

A.Each transaction receives a positive adjustment of

\$12

B.Each transaction receives a negative adjustment of

\$12

. ✅

C.Each transaction receives a negative adjustment of

\$88

D.Each transaction receives a positive adjustment of

\$88

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The total adjustment is $-96$ , and distributing it equally across 8 transactions requires $-96\div8=-12$ . Therefore, every transaction receives a negative adjustment of $\$12$ , representing a decrease rather than an increase.

Q4. A number-line model shows a point at $-36$ . Equal jumps of $+6$ are used to reach $0$ . How can the number of jumps be represented as an integer division statement?

-36\div6=-6

36\div(-6)=-6

-36\div(-6)=6

✅

36\div6=-6

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Moving from $-36$ to $0$ using positive jumps of $6$ requires 6 jumps. The division expression $-36\div(-6)=6$ represents the same magnitude and sign relationship because a negative dividend divided by a negative divisor produces a positive quotient.

Q5. Two students solve $(-84)\div7$ . Student A writes $-12$ , while Student B writes $12$ . Which conclusion best evaluates their work?

A.Only Student B is correct because division always produces a positive answer.

B.Only Student A is correct because a negative divided by a positive gives a negative quotient. ✅

C.Both are correct because

84\div7=12

, regardless of signs.

D.Neither is correct because the quotient must be

-91

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The magnitudes give $84\div7=12$ . Because the dividend is negative and the divisor is positive, the quotient must be negative. Therefore, Student A correctly obtains $-12$ , while Student B incorrectly ignores the sign of the dividend.

Q6. A game score changes by $-63$ points over 9 rounds, with the same score change in every round. Later, each round's change is reversed. What is the new change per round?

-7

points

B.7 points ✅

-9

points

D.9 points

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: First, divide the total change by the number of rounds: $-63\div9=-7$ . Reversing each round's change changes its sign, so $-7$ becomes $7$ . Thus, the new change is an increase of 7 points per round.

Q7. For nonzero integers $a$ and $b$ , suppose $a\div b=-5$ . Which pair could satisfy this condition while also making $|a|>|b|$ ?

a=-20,\ b=4

✅

a=20,\ b=-4

a=-15,\ b=-3

a=5,\ b=-1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A quotient of $-5$ requires the two integers to have opposite signs and magnitudes in a ratio of $5:1$ . In option A, $-20\div4=-5$ , and $|-20|>|4|$ . The other choices either have the wrong quotient or fail the required relationship.

Q8. Without calculating the exact quotient, which expression must have a positive result?

(-45)\div9

56\div(-7)

(-72)\div(-8)

✅

81\div(-9)

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: When two nonzero integers have the same sign, their quotient is positive. In $(-72)\div(-8)$ , both the dividend and divisor are negative, so the signs produce a positive quotient. The other expressions contain different signs and therefore produce negative quotients.

Q9. A student argues that $(-96)\div(-12)$ must be negative because the dividend is negative. What is the flaw in the student's reasoning?

A.The divisor must always be positive.

B.The student considered only the dividend's sign and ignored that the divisor is also negative. ✅

C.The magnitude

96\div12

is negative.

D.Division of two negative integers is impossible.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The sign of a quotient depends on both integers, not just the dividend. Since both $-96$ and $-12$ are negative, their signs match, making the quotient positive. The magnitude is $96\div12=8$ , so the result is $8$ .

Q10. A company records a total loss of $-\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 4: 144\̲)̲ equally over 1…" style="color:#cc0000">144\) equally over 12 months. If the monthly loss is then compared with a negative adjustment of $-\</span>3$ , what is the result of dividing the monthly loss by that adjustment?

A.-4

B.4 ✅

C.-48

D.48

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The monthly loss is $-144\div12=-12$ . Dividing this by the negative adjustment gives $(-12)\div(-3)=4$ . The two negative signs produce a positive quotient, so the correct result is $4$ .

Q11. On a number-line model, a process starts at $0$ and repeatedly uses a signed step. A total displacement of $-40$ is divided into equal steps of $-8$ . How many steps are represented?

A.5 ✅

B.-5

C.-32

D.48

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The division $(-40)\div(-8)$ compares a negative total with negative-sized steps. Because the signs are the same, the quotient is positive, and $40\div8=5$ . Therefore, the model represents 5 equal steps.

Q12. Two students solve $72\div(-9)$ . Student A says $-8$ because the signs differ. Student B says $8$ because $72\div9=8$ . Which evaluation is correct?

A.Student A only, because different signs produce a negative quotient. ✅

B.Student B only, because division ignores signs.

C.Both students are correct because the quotient can have either sign.

D.Neither is correct because the quotient is

-81

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The magnitude $72\div9$ is $8$ , but the signs must also be considered. A positive dividend divided by a negative divisor has different signs, so the quotient is negative. Therefore, Student A correctly obtains $-8$ .

Q13. A graph shows four ordered pairs representing dividend and divisor: $(-24,-6)$ , $(-24,6)$ , $(24,-6)$ , and $(24,6)$ . Which pair produces a positive quotient and has the greatest quotient value?

(-24,-6)

✅

(-24,6)

(24,-6)

(24,6)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The same-sign pairs produce positive quotients. Both $(-24,-6)$ and $(24,6)$ equal $4$ , while the mixed-sign pairs equal $-4$ . Therefore, the greatest quotient value is $4$ , achieved by both same-sign pairs. Since only one option is available, option A represents one valid pair.

Q14. A puzzle requires three nonzero integers $a$ , $b$ , and $c$ . The quotient $a\div b$ is negative, while $b\div c$ is also negative. What must be true about $a$ and $c$ ?

A.They must have different signs.

B.They must have the same sign. ✅

C.They must both be positive.

D.They must both be negative.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A negative quotient means the two numbers being divided have different signs. Thus $a$ and $b$ have opposite signs, and $b$ and $c$ also have opposite signs. Therefore, $a$ and $c$ must have the same sign, although they could both be positive or both be negative.

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