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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

Identify a friendly pair

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

The prompt's addends 25 and 75 make a friendly sum of 100. Addition's commutative property allows the addends to change order without changing their values or the total.

02

Reorder without changing addends

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

Move 75 beside 25 and move 7 to the end. The same three addends—25, 7, and 75—still appear exactly once, so the rewrite is equivalent.

03

Add the friendly pair

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

Evaluate the adjacent friendly pair first. Twenty-five plus seventy-five equals one hundred, leaving the original 7 to add.

04

Find and check the sum

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

The reordered sum is 107. Subtracting the unpaired 7 returns the friendly subtotal 100, confirming the final addition.

05

State the rewrite and value

\[25~+~75~+~7~=~107\]

The efficient order places 25 and 75 together, and the value is 107. This matches choice A.

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Another way

  1. Group the original expression as 25 + (7 + 75) = 25 + 82 = 107; this is valid but less mentally efficient than making 100 first.

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Common mistake

Do not change an addend while reordering. The commutative property moves 7; it does not turn 7 into 70.

Solution walkthrough video