Find an original quantity from a percentage

What you will learn

• Find an original quantity from a percentage

Models used: number sentences, stories, tables.

Before you start

• A percentage of a quantity

Build the whole from the known part

We know 25% is 20. That means one quarter is 20.

1. All the small boxes represent equally sized units, even when they belong to different rows.

   We know 25% is 20. That means one quarter is 20.

   The given amount represents only its stated percentage. Find the value of one equal percentage unit, then rebuild the whole.

   Whole

   4 equal units

2. All the small boxes represent equally sized units, even when they belong to different rows.

   Four quarters make the whole. Put 20 in each equal part.

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Whole2020202080 altogether

   4 equal units · 4 units = 80 · 1 unit = 20

3. The full bar represents 80, which is 100%. The shaded share is 25%.

   The original whole is 80. Check: a quarter of 80 is 20.

   20 × 4 = 80

   Each section: 5% = 4

   0%100% = 80

   25% of 80 = 20

Worked example: A new example, together

25% of a quantity is 15. What is the whole quantity?

1. The whole is unknown and represents 100%. Read each value with the heading above its column.

   Label which amount belongs to the given percentage and which unknown belongs to 100%.

   Scale the known percentage part to one percent, then to one hundred percent.

   The whole is unknown and represents 100%.

   Known share (%)	Amount in that share
   25	15

2. The whole is unknown and represents 100%. Read each value with the heading above its column.

   25 equal percentage units represent 15.

   The whole is unknown and represents 100%.

   Known share (%)	Amount in that share
   25	15

   25% of a quantity is 15. What is the whole quantity?

   60

3. The whole is unknown and represents 100%. Read each value with the heading above its column.

   25 equal percentage units represent 15.

   The whole is unknown and represents 100%.

   Known share (%)	Amount in that share
   25	15

4. The whole is unknown and represents 100%. Read each value with the heading above its column.

   One percent is 15 ÷ 25 = 0.6.

   The whole is unknown and represents 100%.

   Known share (%)	Amount in that share
   25	15

5. The whole is unknown and represents 100%. Read each value with the heading above its column.

   100% is 0.6 × 100 = 60.

   The whole is unknown and represents 100%.

   Known share (%)	Amount in that share
   25	15

6. The whole is unknown and represents 100%. Read each value with the heading above its column.

   Check the result against the question we started with.

   25% of 60 = 15.

   We found 60. Scale the known percentage part to one percent, then to one hundred percent.

   The whole is unknown and represents 100%.

   Known share (%)	Amount in that share
   25	15

Common mistakes to discuss

• Taking the same percentage of the part. Work back to the whole. If 25% is 15, the whole is 15 × 4 = 60; taking 25% of 15 gives 3.75, which answers a different question.

Check your understanding

• Find an original quantity from a percentage
• recover a whole from a known percentage share
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.