← Recover the original number from a wrong calculation · Work Backwards to Recover a Start Value

Recover the original number from a wrong calculation · 12 practice problems

5.NF.A.15.NF.A.2

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 2512\frac{25}{12}

A number was supposed to have 1561\frac{5}{6} subtracted from it, but by mistake 1561\frac{5}{6} was added to it instead, giving 5345\frac{3}{4}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 1561\frac{5}{6} taken away. Instead it was added, and the result came to 5345\frac{3}{4}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 1561\frac{5}{6}.
  • It was added when it should have been subtracted.
  • The wrong answer was 5345\frac{3}{4}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 1561\frac{5}{6} gave 5345\frac{3}{4}, so taking it back off returns the hidden number.
534−1565\frac{3}{4} - 1\frac{5}{6}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 12 and subtract.
534−156=311125\frac{3}{4} - 1\frac{5}{6} = 3\frac{11}{12}
The hidden number is 311123\frac{11}{12}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 1561\frac{5}{6} off the hidden number, as was intended all along.
31112−156=21123\frac{11}{12} - 1\frac{5}{6} = 2\frac{1}{12}
The correct answer is 21122\frac{1}{12}.
Answer: 2512\frac{25}{12}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 5345\frac{3}{4} - 21122\frac{1}{12} should be 2 x 1561\frac{5}{6}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 1561\frac{5}{6}, so the answer is 5345\frac{3}{4} - 3233\frac{2}{3} = 21122\frac{1}{12}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 2 easy answer: 1912\frac{19}{12}

A number was supposed to have 2132\frac{1}{3} subtracted from it, but by mistake 2132\frac{1}{3} was added to it instead, giving 6146\frac{1}{4}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 2132\frac{1}{3} taken away. Instead it was added, and the result came to 6146\frac{1}{4}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 2132\frac{1}{3}.
  • It was added when it should have been subtracted.
  • The wrong answer was 6146\frac{1}{4}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 2132\frac{1}{3} gave 6146\frac{1}{4}, so taking it back off returns the hidden number.
614−2136\frac{1}{4} - 2\frac{1}{3}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 12 and subtract.
614−213=311126\frac{1}{4} - 2\frac{1}{3} = 3\frac{11}{12}
The hidden number is 311123\frac{11}{12}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 2132\frac{1}{3} off the hidden number, as was intended all along.
31112−213=17123\frac{11}{12} - 2\frac{1}{3} = 1\frac{7}{12}
The correct answer is 17121\frac{7}{12}.
Answer: 1912\frac{19}{12}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 6146\frac{1}{4} - 17121\frac{7}{12} should be 2 x 2132\frac{1}{3}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 2132\frac{1}{3}, so the answer is 6146\frac{1}{4} - 4234\frac{2}{3} = 17121\frac{7}{12}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 3 easy answer: 435\frac{4}{35}

A number was supposed to have 3453\frac{4}{5} subtracted from it, but by mistake 3453\frac{4}{5} was added to it instead, giving 7577\frac{5}{7}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 3453\frac{4}{5} taken away. Instead it was added, and the result came to 7577\frac{5}{7}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 3453\frac{4}{5}.
  • It was added when it should have been subtracted.
  • The wrong answer was 7577\frac{5}{7}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 3453\frac{4}{5} gave 7577\frac{5}{7}, so taking it back off returns the hidden number.
757−3457\frac{5}{7} - 3\frac{4}{5}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 35 and subtract.
757−345=332357\frac{5}{7} - 3\frac{4}{5} = 3\frac{32}{35}
The hidden number is 332353\frac{32}{35}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 3453\frac{4}{5} off the hidden number, as was intended all along.
33235−345=4353\frac{32}{35} - 3\frac{4}{5} = \frac{4}{35}
The correct answer is 435\frac{4}{35}.
Answer: 435\frac{4}{35}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 7577\frac{5}{7} - 435\frac{4}{35} should be 2 x 3453\frac{4}{5}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 3453\frac{4}{5}, so the answer is 7577\frac{5}{7} - 7357\frac{3}{5} = 435\frac{4}{35}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 4 easy answer: 73\frac{7}{3}

A number was supposed to have 3143\frac{1}{4} subtracted from it, but by mistake 3143\frac{1}{4} was added to it instead, giving 8568\frac{5}{6}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 3143\frac{1}{4} taken away. Instead it was added, and the result came to 8568\frac{5}{6}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 3143\frac{1}{4}.
  • It was added when it should have been subtracted.
  • The wrong answer was 8568\frac{5}{6}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 3143\frac{1}{4} gave 8568\frac{5}{6}, so taking it back off returns the hidden number.
856−3148\frac{5}{6} - 3\frac{1}{4}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 12 and subtract.
856−314=57128\frac{5}{6} - 3\frac{1}{4} = 5\frac{7}{12}
The hidden number is 57125\frac{7}{12}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 3143\frac{1}{4} off the hidden number, as was intended all along.
5712−314=2135\frac{7}{12} - 3\frac{1}{4} = 2\frac{1}{3}
The correct answer is 2132\frac{1}{3}.
Answer: 73\frac{7}{3}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 8568\frac{5}{6} - 2132\frac{1}{3} should be 2 x 3143\frac{1}{4}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 3143\frac{1}{4}, so the answer is 8568\frac{5}{6} - 6126\frac{1}{2} = 2132\frac{1}{3}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 5 medium answer: 1320\frac{13}{20}

A number was supposed to have 4384\frac{3}{8} subtracted from it, but by mistake 4384\frac{3}{8} was added to it instead, giving 9259\frac{2}{5}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 4384\frac{3}{8} taken away. Instead it was added, and the result came to 9259\frac{2}{5}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 4384\frac{3}{8}.
  • It was added when it should have been subtracted.
  • The wrong answer was 9259\frac{2}{5}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 4384\frac{3}{8} gave 9259\frac{2}{5}, so taking it back off returns the hidden number.
925−4389\frac{2}{5} - 4\frac{3}{8}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 40 and subtract.
925−438=51409\frac{2}{5} - 4\frac{3}{8} = 5\frac{1}{40}
The hidden number is 51405\frac{1}{40}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 4384\frac{3}{8} off the hidden number, as was intended all along.
5140−438=13205\frac{1}{40} - 4\frac{3}{8} = \frac{13}{20}
The correct answer is 1320\frac{13}{20}.
Answer: 1320\frac{13}{20}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 9259\frac{2}{5} - 1320\frac{13}{20} should be 2 x 4384\frac{3}{8}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 4384\frac{3}{8}, so the answer is 9259\frac{2}{5} - 8348\frac{3}{4} = 1320\frac{13}{20}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 6 medium answer: 1718\frac{17}{18}

A number was supposed to have 1791\frac{7}{9} subtracted from it, but by mistake 1791\frac{7}{9} was added to it instead, giving 4124\frac{1}{2}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 1791\frac{7}{9} taken away. Instead it was added, and the result came to 4124\frac{1}{2}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 1791\frac{7}{9}.
  • It was added when it should have been subtracted.
  • The wrong answer was 4124\frac{1}{2}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 1791\frac{7}{9} gave 4124\frac{1}{2}, so taking it back off returns the hidden number.
412−1794\frac{1}{2} - 1\frac{7}{9}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 18 and subtract.
412−179=213184\frac{1}{2} - 1\frac{7}{9} = 2\frac{13}{18}
The hidden number is 213182\frac{13}{18}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 1791\frac{7}{9} off the hidden number, as was intended all along.
21318−179=17182\frac{13}{18} - 1\frac{7}{9} = \frac{17}{18}
The correct answer is 1718\frac{17}{18}.
Answer: 1718\frac{17}{18}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 4124\frac{1}{2} - 1718\frac{17}{18} should be 2 x 1791\frac{7}{9}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 1791\frac{7}{9}, so the answer is 4124\frac{1}{2} - 3593\frac{5}{9} = 1718\frac{17}{18}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 7 medium answer: 12845\frac{128}{45}

A number was supposed to have 23102\frac{3}{10} subtracted from it, but by mistake 23102\frac{3}{10} was added to it instead, giving 7497\frac{4}{9}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 23102\frac{3}{10} taken away. Instead it was added, and the result came to 7497\frac{4}{9}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 23102\frac{3}{10}.
  • It was added when it should have been subtracted.
  • The wrong answer was 7497\frac{4}{9}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 23102\frac{3}{10} gave 7497\frac{4}{9}, so taking it back off returns the hidden number.
749−23107\frac{4}{9} - 2\frac{3}{10}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 90 and subtract.
749−2310=513907\frac{4}{9} - 2\frac{3}{10} = 5\frac{13}{90}
The hidden number is 513905\frac{13}{90}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 23102\frac{3}{10} off the hidden number, as was intended all along.
51390−2310=238455\frac{13}{90} - 2\frac{3}{10} = 2\frac{38}{45}
The correct answer is 238452\frac{38}{45}.
Answer: 12845\frac{128}{45}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 7497\frac{4}{9} - 238452\frac{38}{45} should be 2 x 23102\frac{3}{10}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 23102\frac{3}{10}, so the answer is 7497\frac{4}{9} - 4354\frac{3}{5} = 238452\frac{38}{45}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 8 medium answer: 4924\frac{49}{24}

A number was supposed to have 4164\frac{1}{6} subtracted from it, but by mistake 4164\frac{1}{6} was added to it instead, giving 103810\frac{3}{8}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 4164\frac{1}{6} taken away. Instead it was added, and the result came to 103810\frac{3}{8}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 4164\frac{1}{6}.
  • It was added when it should have been subtracted.
  • The wrong answer was 103810\frac{3}{8}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 4164\frac{1}{6} gave 103810\frac{3}{8}, so taking it back off returns the hidden number.
1038−41610\frac{3}{8} - 4\frac{1}{6}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 24 and subtract.
1038−416=652410\frac{3}{8} - 4\frac{1}{6} = 6\frac{5}{24}
The hidden number is 65246\frac{5}{24}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 4164\frac{1}{6} off the hidden number, as was intended all along.
6524−416=21246\frac{5}{24} - 4\frac{1}{6} = 2\frac{1}{24}
The correct answer is 21242\frac{1}{24}.
Answer: 4924\frac{49}{24}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 103810\frac{3}{8} - 21242\frac{1}{24} should be 2 x 4164\frac{1}{6}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 4164\frac{1}{6}, so the answer is 103810\frac{3}{8} - 8138\frac{1}{3} = 21242\frac{1}{24}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 9 hard answer: 5833\frac{58}{33}

A number was supposed to have 25112\frac{5}{11} subtracted from it, but by mistake 25112\frac{5}{11} was added to it instead, giving 6236\frac{2}{3}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 25112\frac{5}{11} taken away. Instead it was added, and the result came to 6236\frac{2}{3}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 25112\frac{5}{11}.
  • It was added when it should have been subtracted.
  • The wrong answer was 6236\frac{2}{3}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 25112\frac{5}{11} gave 6236\frac{2}{3}, so taking it back off returns the hidden number.
623−25116\frac{2}{3} - 2\frac{5}{11}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 33 and subtract.
623−2511=47336\frac{2}{3} - 2\frac{5}{11} = 4\frac{7}{33}
The hidden number is 47334\frac{7}{33}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 25112\frac{5}{11} off the hidden number, as was intended all along.
4733−2511=125334\frac{7}{33} - 2\frac{5}{11} = 1\frac{25}{33}
The correct answer is 125331\frac{25}{33}.
Answer: 5833\frac{58}{33}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 6236\frac{2}{3} - 125331\frac{25}{33} should be 2 x 25112\frac{5}{11}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 25112\frac{5}{11}, so the answer is 6236\frac{2}{3} - 410114\frac{10}{11} = 125331\frac{25}{33}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 10 hard answer: 3721\frac{37}{21}

A number was supposed to have 5275\frac{2}{7} subtracted from it, but by mistake 5275\frac{2}{7} was added to it instead, giving 121312\frac{1}{3}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 5275\frac{2}{7} taken away. Instead it was added, and the result came to 121312\frac{1}{3}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 5275\frac{2}{7}.
  • It was added when it should have been subtracted.
  • The wrong answer was 121312\frac{1}{3}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 5275\frac{2}{7} gave 121312\frac{1}{3}, so taking it back off returns the hidden number.
1213−52712\frac{1}{3} - 5\frac{2}{7}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 21 and subtract.
1213−527=712112\frac{1}{3} - 5\frac{2}{7} = 7\frac{1}{21}
The hidden number is 71217\frac{1}{21}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 5275\frac{2}{7} off the hidden number, as was intended all along.
7121−527=116217\frac{1}{21} - 5\frac{2}{7} = 1\frac{16}{21}
The correct answer is 116211\frac{16}{21}.
Answer: 3721\frac{37}{21}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 121312\frac{1}{3} - 116211\frac{16}{21} should be 2 x 5275\frac{2}{7}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 5275\frac{2}{7}, so the answer is 121312\frac{1}{3} - 104710\frac{4}{7} = 116211\frac{16}{21}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 11 hard answer: 6142\frac{61}{42}

A number was supposed to have 65126\frac{5}{12} subtracted from it, but by mistake 65126\frac{5}{12} was added to it instead, giving 142714\frac{2}{7}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 65126\frac{5}{12} taken away. Instead it was added, and the result came to 142714\frac{2}{7}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 65126\frac{5}{12}.
  • It was added when it should have been subtracted.
  • The wrong answer was 142714\frac{2}{7}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 65126\frac{5}{12} gave 142714\frac{2}{7}, so taking it back off returns the hidden number.
1427−651214\frac{2}{7} - 6\frac{5}{12}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 84 and subtract.
1427−6512=7738414\frac{2}{7} - 6\frac{5}{12} = 7\frac{73}{84}
The hidden number is 773847\frac{73}{84}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 65126\frac{5}{12} off the hidden number, as was intended all along.
77384−6512=119427\frac{73}{84} - 6\frac{5}{12} = 1\frac{19}{42}
The correct answer is 119421\frac{19}{42}.
Answer: 6142\frac{61}{42}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 142714\frac{2}{7} - 119421\frac{19}{42} should be 2 x 65126\frac{5}{12}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 65126\frac{5}{12}, so the answer is 142714\frac{2}{7} - 125612\frac{5}{6} = 119421\frac{19}{42}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.
Variant 12 hard answer: 2120\frac{21}{20}

A number was supposed to have 7357\frac{3}{5} subtracted from it, but by mistake 7357\frac{3}{5} was added to it instead, giving 161416\frac{1}{4}. Find the value of the correct calculation.

Show solution
1 · Understandwhat's really being asked

A hidden number should have had 7357\frac{3}{5} taken away. Instead it was added, and the result came to 161416\frac{1}{4}. We need what the correct subtraction would have given.

Givens
  • The amount involved is 7357\frac{3}{5}.
  • It was added when it should have been subtracted.
  • The wrong answer was 161416\frac{1}{4}.
Unknowns
  • The value of the correct calculation.
Constraints
  • The hidden number is the same in both the wrong and the right calculation.
2 · Planchoose the strategy

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

The answer cannot be reached from the wrong result in one step. Undo the mistaken addition to recover the hidden number first, then do the subtraction that was meant.

3 · Execute3 carry out the plan

1Undo the mistaken addition

#11 Work Backwards 5.NF.A.2
Adding 7357\frac{3}{5} gave 161416\frac{1}{4}, so taking it back off returns the hidden number.
1614−73516\frac{1}{4} - 7\frac{3}{5}
Subtraction undoes the addition that should not have happened.

2Find the hidden number with a common denominator

#7 Identify Subproblems 5.NF.A.1
Put both over 20 and subtract.
1614−735=8132016\frac{1}{4} - 7\frac{3}{5} = 8\frac{13}{20}
The hidden number is 813208\frac{13}{20}.

3Do the correct calculation (subtract)

#7 Identify Subproblems 5.NF.A.1
Now take 7357\frac{3}{5} off the hidden number, as was intended all along.
81320−735=11208\frac{13}{20} - 7\frac{3}{5} = 1\frac{1}{20}
The correct answer is 11201\frac{1}{20}.
Answer: 2120\frac{21}{20}
4 · Reviewdoes it hold up?

The wrong answer and the right one differ by twice the amount involved: 161416\frac{1}{4} - 11201\frac{1}{20} should be 2 x 7357\frac{3}{5}, and it is.

Another way: Skip the hidden number entirely: subtracting instead of adding moves the result down by two lots of 7357\frac{3}{5}, so the answer is 161416\frac{1}{4} - 151515\frac{1}{5} = 11201\frac{1}{20}.

Standardsmin grade 5
  • 5.NF.A.1 Add and subtract fractions with unlike denominators — Both subtractions, over unlike denominators.
  • 5.NF.A.2 Solve word problems involving addition and subtraction of fractions — Reading the mistake as an addition to be undone.
💡Takeaway. A wrong answer still holds the hidden number. Undo the mistake first, then do what was actually asked.