Reverse decimal operations to recover the start

5.NBT.B.7 · take · grade 5

Archetype: Work Backwards to Recover a Start Value · step in a 9-type progression

▶ Practice — 10 problems

A certain number's $\frac{1}{100}$ is $0.007$ less than $65.42$ . Find the number.

Show solution

Understand

Some number divided by 100 (its one-hundredth) equals a value that is 0.007 less than 65.42. Find the original number.

Givens

(the number) x 1/100 = 65.42 - 0.007.
1/100 of a number means the number divided by 100.

Unknowns

The original number.

Constraints

0.007 is in the thousandths place, so the right side is a thousandths decimal.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

First compute the value that equals one-hundredth of the number, then undo the 'divide by 100' by multiplying by 100 to recover the original number.

Execute

#11 Work Backwards 5.NBT.B.7

The number's hundredth is 0.007 less than 65.42, so subtract: 65.42 - 0.007 = 65.413.

65.42 - 0.007 = 65.413

'0.007 less than' means subtract 0.007.

#11 Work Backwards 5.NBT.B.7

If 1/100 of the number is 65.413, then the number itself is 100 times that. Multiplying by 100 shifts the decimal point two places right: 65.413 x 100 = 6541.3.

65.413 \times 100 = 6541.3

Taking 1/100 shrank the number, so multiplying by 100 grows it back.

Answer: 6541.3

Review

Check forward: 6541.3 / 100 = 65.413, and 65.413 is 0.007 less than 65.42 (65.42 - 65.413 = 0.007). Everything matches.

Guess and Check the place-value shift: since 1/100 of the answer must be about 65.4, the answer is about 6540; refining to 6541.3 makes the hundredth exactly 65.413.

Standards · min grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting 65.42 - 0.007 and multiplying 65.413 x 100.
5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Shifting the decimal point two places when multiplying by 100.

💡 '1/100 of it is...' just means divide by 100 - so multiply by 100 to get the number back!