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M.1 Estimate sums and differences of mixed numbers

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What does it mean to estimate sums and differences of mixed numbers?

Estimating sums and differences of mixed numbers means finding an approximate answer rather than the exact one. You round each mixed number to a whole number or an easy fraction before adding or subtracting.

Example:
  • \( 3 \tfrac{2}{3} + 4 \tfrac{1}{4} \approx 4 + 4 = 8 \)
Note

Estimation helps you decide whether your exact answer makes sense. It gives you a quick way to predict or check your work.

How to estimate sums of mixed numbers

To estimate the sum of mixed numbers, round each number to the nearest whole number, then add them together.

Example:
  • Problem: \( 5 \tfrac{3}{8} + 2 \tfrac{5}{6} \)
  • Round \( 5 \tfrac{3}{8} \) to \( 5 \) (since \( \tfrac{3}{8} < \tfrac{1}{2} \))
  • Round \( 2 \tfrac{5}{6} \) to \( 3 \) (since \( \tfrac{5}{6} > \tfrac{1}{2} \))
  • Add: \( 5 + 3 = 8 \)
  • Estimated sum: \( \approx 8 \)
Note

When rounding, fractions greater than or equal to \( \tfrac{1}{2} \) round up to the next whole number. Fractions less than \( \tfrac{1}{2} \) round down.

How to estimate differences of mixed numbers

To estimate the difference of mixed numbers, round each mixed number to the nearest whole number, then subtract the smaller from the larger.

Example:
  • Problem: \( 7 \tfrac{5}{8} - 2 \tfrac{1}{3} \)
  • Round \( 7 \tfrac{5}{8} \) to \( 8 \) (since \( \tfrac{5}{8} > \tfrac{1}{2} \))
  • Round \( 2 \tfrac{1}{3} \) to \( 2 \) (since \( \tfrac{1}{3} < \tfrac{1}{2} \))
  • Subtract: \( 8 - 2 = 6 \)
  • Estimated difference: \( \approx 6 \)
Note

Estimation gives a quick check to see if your exact subtraction is reasonable. Always compare your estimated and actual results.

Why estimating is useful

Estimation helps you make quick and reasonable decisions without detailed calculations. It is useful in real-life situations such as measuring, shopping, or comparing quantities.

Example in real life:
  • You need two boards: one is \( 4 \tfrac{3}{4} \) feet long and another is \( 2 \tfrac{2}{5} \) feet long. You estimate \( 5 + 2 = 7 \) feet. You can plan to buy about 7 feet of material.
Note

Estimation helps you check for reasonableness and make faster mental calculations in everyday tasks.

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