Numbers & Place Value

Depth · one concept from first principles · step 2 of 4

Find the number before it was rounded

A number was divided by 77, and the quotient rounded to the nearest whole number came to 1313. How many natural numbers could that number be?

ExpressionsNumber systemBase-ten numbers
Your answer
How to solve
Strategy Work Backwards — Rounding throws information away, so work backwards to the range it came from rather than to a single value. Multiply that range by 7 and count the whole numbers inside.
1STEP 1

Turn the rounded value back into a range

A quotient rounds to 13 exactly when it is at least 12.5 and below 13.5.

12.5 ≤ (quotient) ＜ 13.5
2STEP 2

Multiply the range by 7

The quotient is the number divided by 7, so the number itself is 7 times each boundary.

12.5 × 7 = 87.5, 13.5 × 7 = 94.5
3STEP 3

Count the natural numbers inside

The whole numbers strictly between 87.5 and 94.5 run from 88 to 94, and a run of whole numbers counts as last minus first plus one.

94 - 88 + 1 = 7
Answer
7
Check the ends: 88 ÷ 7 ≈ 12.57 rounds to 13, and 94 ÷ 7 ≈ 13.43 rounds to 13 too, while 87 and 95 fall outside.
Takeaway

A rounded answer came from a range, not a single number -- work back to the range.

  • Turn the rounded value back into a range
  • Multiply the range by 7
  • Count the natural numbers inside
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Overlap of two ranges on a number lineInclusive vs exclusive range boundaries
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▶ Practice — 12 problems

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