Undo a wrong fraction operation to recover the start

4.NF.B.3 · take · grade 4

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

You were supposed to subtract $\dfrac{4}{11}$ from a number, but by mistake you added it instead, and the result was $\dfrac{10}{11}$ . Find the value you would get from the correct calculation.

Show solution

Understand

You were supposed to subtract 4/11 from a number, but accidentally added 4/11 and got 10/11. I must first recover the original number, then do the correct subtraction.

Givens

The intended operation: number minus 4/11.
The mistaken operation: number plus 4/11, which gave 10/11.
All fractions have denominator 11.

Unknowns

The original number.
The result of the correct calculation (number minus 4/11).

Constraints

Same denominator 11 throughout, so adding/subtracting just works on numerators.

Plan

#11 Work Backwards · also uses: #7 Identify Subproblems

The end result of the wrong calculation is given, so work backwards: undo the mistaken addition to recover the original number, then perform the correct subtraction.

Execute

#11 Work Backwards 4.NF.B.3

The mistake added 4/11 to the number to get 10/11. Undo it by subtracting 4/11: original number = 10/11 - 4/11 = 6/11.

\dfrac{10}{11}-\dfrac{4}{11}=\dfrac{6}{11}

Working backwards undoes the wrong step by doing its opposite.

#7 Identify Subproblems 4.NF.B.3

The correct operation subtracts 4/11 from the original number: 6/11 - 4/11 = 2/11.

\dfrac{6}{11}-\dfrac{4}{11}=\dfrac{2}{11}

With the real number found, the intended subtraction is a one-step like-denominator subtraction.

Answer: 2/11

Review

Check: original 6/11 plus 4/11 really is 10/11 (matches the mistake), and 6/11 minus 4/11 is 2/11. The correct answer 2/11 is smaller than the wrong result 10/11, which makes sense since subtracting should give less than adding.

Use the shortcut (tool 5): the wrong result is too big by 2 × 4/11 = 8/11, so the correct answer = 10/11 - 8/11 = 2/11.

Standards · min grade 4

4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting like-denominator elevenths to recover the number and compute the correct result.

💡 This only needs Grade 4 fraction subtraction — work backwards to undo the mistake, then subtract the right way!