Eliminate to find two decimals from sum and difference

5.NBT.B.7 · take · grade 5

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

▶ Practice — 10 problems

Two decimal numbers have a sum of $12.46$ and a difference of $5.68$ . Find each of the two numbers.

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Understand

Two decimal numbers add up to 12.46 and differ by 5.68. Find both numbers.

Givens

The two numbers add to 12.46.
The larger minus the smaller is 5.68.

Unknowns

The larger number and the smaller number.

Constraints

Both are decimals; the larger one is bigger than the smaller by 5.68.

Plan

#1 Draw a Diagram · also uses: #15 Organize Information in More Ways

Picture the two numbers as two bars. The sum is both bars together; the difference is the extra piece on the longer bar. Adding sum and difference gives two copies of the larger number, which is the classic sum-and-difference (bar model) method.

Execute

#1 Draw a Diagram 5.NBT.B.7

In a bar model, the longer bar plus the shorter bar is 12.46, and the longer minus the shorter is 5.68. Adding these removes the shorter bar twice over and leaves two longer bars: 12.46 + 5.68 = 18.14 = two larger numbers.

12.46 + 5.68 = 18.14 = 2 \times (\text{larger})

Sum plus difference doubles the bigger number - a bar-model staple.

#1 Draw a Diagram 5.NBT.B.7

Half of 18.14 is the larger number: 18.14 / 2 = 9.07.

18.14 \div 2 = 9.07

Two equal larger bars share 18.14 evenly, so split it in half.

#15 Organize Information in More Ways 5.NBT.B.7

Subtract the larger from the total to get the smaller: 12.46 - 9.07 = 3.39. (Equivalently, 9.07 - 5.68 = 3.39.)

12.46 - 9.07 = 3.39

Once one number is known, the other is whatever is left of the sum.

Answer: 9.07 and 3.39

Review

Check both clues: 9.07 + 3.39 = 12.46 (sum correct) and 9.07 - 3.39 = 5.68 (difference correct). Both match the givens.

Subtract instead: sum minus difference, 12.46 - 5.68 = 6.78, equals two smaller numbers, so the smaller is 6.78 / 2 = 3.39 and the larger is 3.39 + 5.68 = 9.07 - same pair.

Standards · min grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Computing 12.46 + 5.68, halving 18.14, and subtracting 12.46 - 9.07.

💡 Sum plus difference is two of the bigger number - cut it in half and the rest falls out!