Use operation properties to make rational-number calculations efficient.
Answer Key
Choose a Shortcut
Use operation properties to simplify each calculation. Record any intermediate steps on separate paper.
1.
The theater club is cutting trim for its set. Use the table and group factors to find the total length of trim.
Set Trim Plan
Set sections
Strips per section
Meters per strip
7
4
0.25
7 × 4 × 0.25 = 7 m
2.
A ticket spreadsheet contains (−1.25) × 13 × 8. Which rewrite uses the commutative and associative properties to create a whole-number factor first?
●A.Use [(−1.25) × 8] × 13.
○B.Use [(−1.25) × 13] + 8.
○C.Use [(−1.25) × (−8)] × 13.
○D.Use [(−1.25) + 8] × 13.
3.
A refund calculation is (−3/4) × 28. Use 28 = 20 + 8 and the distributive property.
(−3/4) × 28 = -21
4.
A set-planning calculation is (−7/8) ÷ (1/4). Which product has the same value?
○A.Multiply (7/8) × 4.
○B.Multiply (−8/7) × (1/4).
●C.Multiply (−7/8) × 4.
○D.Multiply (−7/8) × (1/4).
5.
The club's account contains the calculation (−18 + 4.5) ÷ 1.5. Divide each term in the numerator by 1.5, then add.
(−18 + 4.5) ÷ 1.5 = -9
Build Sets and Sell Seats
Choose an efficient strategy for each situation. Negative amounts represent decreases in the club's account or backward movements.
6.
Each of 18 backdrop frames needs one board costing $2.75 and one connector costing $1.25. Use the distributive property to find the total cost of these materials.
Total cost: $72
7.
The club refunds 12 tickets from one performance and 8 from another. Each refund changes the account by −$6.50. Which expression correctly combines the refunds?
○A.Use (−6.50) × (12 − 8).
○B.Use (−6.50) ÷ (12 + 8).
○C.Use (−6.50) + (12 × 8).
●D.Use (−6.50) × (12 + 8).
8.
A sliding set platform moves backward 0.125 meter with each turn of a handle. Backward movement is negative. During each of 9 checks, the handle makes 8 turns. Regroup the factors to find the total signed movement.
Total signed movement: -9 m
9.
The club sells 49 seats at $7.50 each. Which calculation correctly uses 49 = 50 − 1 to find the revenue?
○A.Calculate 7.50 × 50 − 1.
●B.Calculate 7.50 × 50 − 7.50 × 1.
○C.Calculate 7.50 × 50 + 7.50 × 1.
○D.Calculate (7.50 − 1) × 50.
10.
A crew has 10.5 meters of fabric from one roll and 7.5 meters from another. Each curtain panel needs 0.75 meter. There is no waste. Use (10.5 + 7.5) ÷ 0.75 to find how many panels the crew can make.
The crew can make 24 panels.
Check the Reasoning
Decide which rewrites preserve a value. Pay attention to grouping, signs, and reciprocals.
11.
A crew member claims that (−24 ÷ 3) ÷ 2 equals −24 ÷ (3 ÷ 2) because division is associative. Which statement correctly checks the claim?
●A.The claim is false; the values are −4 and −16, respectively.
○B.The claim is false; the values are −16 and −4, respectively.
○C.The claim is true; both expressions equal −4.
○D.The claim is true; both expressions equal −16.
12.
A planning sheet shows (−5/6) × (6/5) × 2.4. Select every rewrite that preserves its value.
A spreadsheet divides a signed adjustment by −2/3. Complete an equivalent multiplication calculation by writing the reciprocal.
5.4 ÷ (−2/3) = 5.4 × -3/2
14.
Noor needs to evaluate 15 ÷ (0.5 + 0.25) while planning fabric cuts. She rewrites it as (15 ÷ 0.5) + (15 ÷ 0.25). Is her rewrite valid? Find the value of each expression and explain why the distributive property does or does not apply.
The rewrite is not valid. The original expression is 15 ÷ 0.75 = 20. Noor's rewrite is 30 + 60 = 90. Division can be distributed over a sum in the numerator with a common nonzero divisor, but not over a sum in the divisor.