Depth · one concept from first principles · step 2 of 4
Reverse decimal operations to recover the start
A certain number's is less than . Find the number.
Base-ten numbers
Your answer
Watch out It's tempting to report 65.413 as the final answer, but that's only one-hundredth of the number - the number itself is 100 times that.
How to solve
Strategy Work Backwards — 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.
1STEP 1
Find one-hundredth of the number
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.
5.NBT.B.7Work Backwards2STEP 2
Undo the divide-by-100
of the number is 65.413, so the number is 100 times that: 65.413 x 100 = 6541.3.
65.413 × 100 = 6541.3
Taking shrank the number, so multiplying by 100 grows it back.
5.NBT.B.7Work BackwardsSince of the number is 65.413, the number itself is 65.413 x 100 = 6541.3.
Why?
Taking of the number means dividing it by 100, so to recover the number we must undo that division.
Why?
Multiplying by 100 and dividing by 100 reverse each other, so multiplying by 100 cancels the earlier divide-by-100 and hands the number back.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Multiplying 65.413 by 100 slides the decimal point two places to the right, which is why the result reads 6541.3.
Why?
Multiplying by 100 makes each digit worth one hundred times as much, so every digit climbs to a place two steps higher.
Why?
One hundred is ten tens, and each place holds ten of the place just below it, so growing a hundredfold moves up exactly two places.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Answer
6541.3
65.413 × 100 = 6541.3
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.
Takeaway
' of it is...' just means divide by 100 - so multiply by 100 to get the number back!
- Find one-hundredth of the number
- Undo the divide-by-100
Where next?
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →