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All courses Math Foundations · MF.NS.4 4 of 60
Use commutative, associative, and distributive structure to rewrite and simplify numerical expressions efficiently.

Use commutative structure to reorder

Problem
Rewrite 25 + 7 + 75 in an order that makes it easier to add. Then find the value.
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Hint

Look for the two addends that make a friendly number, then place those two next to each other.

Use the commutative property of addition: you can change the order of addends without changing the sum.

Solution walkthrough

01

Spot the easy pair

\[25~+~7~+~75\]

This expression uses addition, so the commutative property lets us change the order of the addends without changing the sum. Here, 25 and 75 make a friendly pair because they add to 100.

02

Rewrite in a friendlier order

\[25~+~7~+~75~=~25~+~75~+~7\]

Only the order changed. The addends are still 25, 7, and 75, so the new expression is equivalent to the original one.

03

Add the friendly pair first

\[25~+~75~+~7~=~100~+~7\]

Putting 25 and 75 together makes the calculation easier because 25 + 75 is exactly 100.

04

Find the value

\[100~+~7~=~107\]

So the rewritten expression has a value of 107, and the original expression 25 + 7 + 75 also equals 107.

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Another way

  1. Check in the original order: 25 + 7 = 32, and 32 + 75 = 107. The sum matches.

!

Common mistake

A common mistake is to change the numbers instead of just their order, such as turning 25 + 7 + 75 into 25 + 70 + 75. The commutative property only changes order; it does not change any addend.