📝 Multiply and Divide Integers in algebraic expressions (7 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 7 questions available

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What is Multiply and Divide Integers in algebraic expressions?

Definition:
Multiplying and dividing integers in algebraic expressions involves performing these operations on signed numbers, where the product or quotient of two integers with the same sign is positive, and with different signs is negative, and these rules extend to coefficients and constants in expressions.

Working:
For multiplication, multiply absolute values and determine sign: positive if both numbers have same sign, negative if different; for division, divide absolute values and apply same sign rule; in expressions, multiply/divide coefficients and constants, then combine with variables.

Example:
Simplify $(-4x) \times 3$ and $(-12) \div (-3)$ .
Solution: $(-4x) \times 3 = -12x$ (different signs, product negative); $(-12) \div (-3) = 4$ (same signs, quotient positive).

Reason:
These operations are essential for solving equations with multiple terms, simplifying rational expressions, and modeling proportional relationships, ensuring correct handling of signed quantities in algebra and applied fields.

Easy

Medium

Hard

📝 All Multiply and Divide Integers in algebraic expressions MCQs

Q1. Which statement correctly predicts the sign of $(-8)\div(-2)$ before calculating its value?

A.The quotient is negative because both numbers are negative.

B.The quotient is positive because two negative signs cancel. ✅

C.The quotient is zero because the signs are opposite.

D.The quotient is negative because division always produces a smaller number.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: When two integers with the same sign are divided, the quotient is positive. Here both numbers are negative, so their signs cancel and the magnitude is $8\div2=4$ , giving $+4$ .

Q2. A student calculates $(-6)(4)(-2)$ by first multiplying $(-6)(4)=-24$ , then claims the final answer must remain negative because multiplying by another negative cannot change the sign. What is the correct conclusion?

A.The answer is

-48

because the first product determines the final sign.

B.The answer is

48

because two negative factors produce a positive overall sign. ✅

C.The answer is

12

because the two negative factors cancel completely.

D.The answer is

-12

because three factors always produce a negative result.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The expression contains two negative factors, $-6$ and $-2$ . Their negative signs combine to produce a positive sign, while the magnitudes multiply as $6\times4\times2=48$ .

Q3. A submarine changes depth by $-12$ meters every 3 minutes. If this rate continues for 15 minutes, which expression and result correctly represent its total change in depth?

15\div(-12)=-1.25

meters

(-12)\div3\times15=-60

meters ✅

(-12)\times3\div15=-2.4

meters

12\div3\times15=60

meters

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The rate is $-12\div3=-4$ meters per minute. Over 15 minutes, the total change is $-4\times15=-60$ meters. The negative sign indicates movement to a greater depth.

Q4. A student solves $(-84)\div7=-12$ , then argues that $84\div(-7)$ must equal $12$ because the negative sign moved from the dividend to the divisor. Which evaluation best identifies the error?

A.The student is correct because moving a sign changes nothing.

B.The student is wrong because changing exactly one factor's sign changes the quotient's sign. ✅

C.The student is wrong because division of integers is always positive.

D.The student is correct because both expressions have the same absolute value.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The magnitude remains $12$ , but the sign depends on whether the two integers have the same or different signs. Since $84$ and $-7$ have different signs, $84\div(-7)=-12$ .

Q5. On a number line, a point starts at $0$ . A rule sends it 3 units in the negative direction for each step, and the rule is applied 4 times. Which final position is consistent with this repeated change?

A.-12 ✅

B.-7

C.12

D.7

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Each application changes the position by $-3$ . Applying the same change four times gives $4\times(-3)=-12$ , so the point ends at $-12$ . The negative direction is preserved through the repeated changes.

Q6. Two methods are proposed for finding $(-72)\div(-6)$ . Method A changes both signs to positive and computes $72\div6$ . Method B computes $72\div6$ but keeps one negative sign because the original numbers were negative. Which comparison is correct?

A.Both methods give

-12

because negative numbers always produce negative quotients.

B.Method A gives

12

, while Method B gives

-12

; Method A is correct because matching signs produce a positive quotient. ✅

C.Method B gives

12

, while Method A gives

-12

; Method B is correct because division preserves one negative sign.

D.Both methods give

12

, but only Method B explains why the result is positive.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For division, integers with the same sign produce a positive quotient. Method A correctly recognizes that both negative signs combine to give a positive result, so $72\div6=12$ .

Q7. A puzzle uses three integers $a$ , $b$ , and $c$ . Their product is negative, and $a\div b$ is positive. Which conclusion must be true about the signs of the three integers?

A.Exactly one of

a,b,c

is negative.

B.Exactly two of

a,b,c

are negative. ✅

C.All three integers are negative.

D.The signs cannot be determined from the information given.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A positive quotient $a\div b$ means $a$ and $b$ have the same sign. Their product is therefore positive. Since the total product $abc$ is negative, $c$ must be negative, making exactly two negatives overall.

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