Find two lengths from sum and difference

2.MD.B.5 · adapt · grade 2

Archetype: Find Two Unknowns from Sum and Difference · step in a 8-type progression

There are two bars, ㉠ and ㉡. The sum of the lengths of ㉠ and ㉡ is $90\ \text{cm}$ , and their difference is $10\ \text{cm}$ . (㉠ is longer than ㉡.)

Find the length, in centimeters, of the longer bar ㉠ and the shorter bar ㉡.

Show solution

Understand

Two bars, A and B, together measure 90 cm. A is 10 cm longer than B. Find each bar's length in centimeters.

Givens

The sum of the two bars' lengths is 90 cm.
The difference between the bars' lengths is 10 cm (A is longer than B).
The figure shows bar A as the full 90 cm length with a 10 cm extra piece on its right end, and bar B as the shorter bar.

Unknowns

The length of the longer bar A in centimeters.
The length of the shorter bar B in centimeters.

Constraints

Both lengths are positive whole numbers of centimeters.
A is longer than B.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture of the two stacked bars lets us see that if we remove the 10 cm extra piece from the total, the remaining length is split into two equal bars (the length of B twice). Picturing the bars turns the sum-and-difference relationship into a simple take-away-then-halve calculation.

Execute

#1 Draw a Diagram 2.MD.B.5

If we cut off the 10 cm extra piece that makes A longer than B, the two bars become the same length. The total drops from 90 cm by 10 cm.

90 - 10 = 80

Taking the extra piece away leaves two equal bars, so the leftover 80 cm is just B counted twice.

#7 Identify Subproblems 2.MD.B.5

The 80 cm is two equal bars (B and a copy of B). Halving it gives the shorter bar B.

80 \div 2 = 40

Sharing 80 equally into two groups is a second-grade halving fact.

#7 Identify Subproblems 2.MD.B.5

The longer bar A is the shorter bar B plus the 10 cm extra piece.

40 + 10 = 50

A is just B with its extra piece restored.

Answer: The longer bar A is 50 cm and the shorter bar B is 40 cm.

Review

Check both facts: 50 + 40 = 90 cm (the sum) and 50 - 40 = 10 cm (the difference). Both match, and A (50) is longer than B (40), so the answer is sensible.

Guess and check (tool 6): try A = 50, B = 40; their sum is 90 and difference is 10, confirming the answer on the first reasonable guess.

Standards · min grade 2

2.MD.B.5 Solve word problems involving lengths using same units — Combining and comparing the two bar lengths (sum and difference) in centimeters to find each length.

💡 Take the extra piece off, split what's left in half, then add the piece back: only Grade 2 add-subtract-and-halve sense!