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Grade 7 · lesson 3 of 4

Recover an Original Amount

A reduced amount is the surviving percent of an unknown original.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: After a 25% reduction, a fictional fee is $30. What was the original fee? Candidate answers: (A) 22.5 USD (B) 37.5 USD (C) 40 USD (D) 55 USD.
  • Independent check inputs: After a 25% reduction, a fictional fee is $45. What was the original fee? Candidate answers: (A) 33.75 USD (B) 56.25 USD (C) 60 USD (D) 70 USD.

Learn the idea

A percent is a ratio per hundred, but its base quantity matters. Successive percent operations act on successive amounts; reverse problems recover the original base rather than simply adding back a percentage. A reduced amount is the surviving percent of an unknown original. Write a multiplication equation and divide by the retained decimal to reverse that relationship accurately.

Here is the method on the supplied teaching case. The remaining fee is 75% of the original. Solve 0.75p=30 by dividing by 0.75: p=40. Adding 25% of the reduced fee would use the wrong base.

Adding the discount rate to the reduced amount uses the wrong base. Check the recovered original by applying the stated reduction and comparing it with the supplied final amount.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: After a 25% reduction, a fictional fee is $30. What was the original fee? Candidate answers: (A) 22.5 USD (B) 37.5 USD (C) 40 USD (D) 55 USD. Solution: The remaining fee is 75% of the original. Solve 0.75p=30 by dividing by 0.75: p=40. Adding 25% of the reduced fee would use the wrong base. The final answer is 40 USD. Adding the discount rate to the reduced amount uses the wrong base. Check the recovered original by applying the stated reduction and comparing it with the supplied final amount.

Your turn

Check your reasoning

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Helpful tips
  • A reduced amount is the surviving percent of an unknown original.
  • Adding the discount rate to the reduced amount uses the wrong base.
After a 25% reduction, a fictional fee is $45. What was the original fee?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

60 USD

The remaining fee is 75% of the original. Solve 0.75p=45 by dividing by 0.75: p=60. Adding 25% of the reduced fee would use the wrong base.

Take it further

Changed-input task: After a 25% reduction, a fictional fee is $135. What was the original fee? Candidate answers: (A) 101.25 USD (B) 160 USD (C) 168.75 USD (D) 180 USD. Show the mathematical reason, not only the selected answer. Expected result: 180 USD. The remaining fee is 75% of the original. Solve 0.75p=135 by dividing by 0.75: p=180. Adding 25% of the reduced fee would use the wrong base.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. After a 25% reduction, a fictional fee is $60. What was the original fee?
Need a hint?
  • A reduced amount is the surviving percent of an unknown original.
  • Adding the discount rate to the reduced amount uses the wrong base.
2. After a 25% reduction, a fictional fee is $75. What was the original fee?
Need a hint?
  • A reduced amount is the surviving percent of an unknown original.
  • Adding the discount rate to the reduced amount uses the wrong base.
3. After a 25% reduction, a fictional fee is $90. What was the original fee?
Need a hint?
  • A reduced amount is the surviving percent of an unknown original.
  • Adding the discount rate to the reduced amount uses the wrong base.

Your choices are temporary. Feedback appears after you check all the answers.

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