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Unit 4 · Standard 6.AT.4

Determine the Whole Given the Part and Percent

Lesson 4-5 · Learn It — how the math works

Why we're learning this

Every percent problem has three pieces: the part, the whole, and the percent. So far the whole has always been handed to you. Today it is the missing piece — a sale price without the original price, a car's current value without its sticker price. Your job is to work backwards to the whole. For example, Akela paid $42 for a sweater on sale, and that was 60% of the original price. What was the original price?

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Working backwards to the whole

When you know a part and the percent it represents, the whole is missing. An equivalent ratio table keeps the percent and amount paired as you scale to 100%. A number line and an equation can show the same relationship.

Picture itSee what the math looks like before you read about it

Three hundred middle school students voted for one of three snacks to add to the cafeteria menu, and the graph shows the PERCENT of students who chose each snack. Today the question runs the other way: if you knew only how many students picked one snack and what percent that was, could you work back to the total?
Where you see this in real lifeThree hundred middle school students voted for one of three snacks to add to the cafeteria menu, and the graph shows the PERCENT of students who chose each snack. Today the question runs the other way: if you knew only how many students picked one snack and what percent that was, could you work back to the total?

Watch me solve itI do — read one step at a time

The problem looks like this
Equivalent ratio table: 60 percent matches 42 dollars; divide both values by 6 to get 10 percent and 7 dollars; multiply both by 10 to get 100 percent and 70 dollars.Percent60%10%100%Price$42$7$70Ratio-table stepKnown partDivide both by 6Multiply both by 10
  1. Akela paid $42 for a sweater on sale, and that was 60% of the original price. The $42 is the ; the original price is the , which is 100%.
  2. I put the matching values in a ratio table: 60% pairs with $42. I want a friendly 10% row, so I divide both values by 6: 10% pairs with $7.
  3. To reach the , I multiply both values in the 10% row by 10: 100% pairs with $70. The original price is $70, which is larger than the sale price.
  4. I can check the table with an : 0.6 × v = 42, so v = 42 ÷ 0.6 = 70. The equation describes the same -to- relationship.

Watch out — mistakes students make here

  • Multiplying by 0.6 makes the number smaller, so the original price would be less than the sale price — impossible for a discount.
  • 60 is the percent, not a number of dollars.
  • Percent problems scale the amount; the part and the percent are never added.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Sheng's car is now worth $26,600, which is 70% of what his parents paid for it. Find the original price.
  2. Name the pieces first: $26,600 is the , 70% is the , and the purchase price is the unknown .
  3. Make a ratio table. Divide both values by 7 to find 10%:$26,600 ÷ 7 = $3,800
  4. Multiply both values by 10 to reach 100%: $3,800 × 10 = $38,000. The purchase price was $38,000.$3,800 × 10 = $38,000

On your ownYou do — solve this one by yourself

Before you start, remember:

  • A student spends 75 minutes in science class, which is 15% of the school day.
  • Which number is the , which is the , and what is the unknown ?
  • Start a ratio table with 15% paired with 75 minutes. Find a friendly row, then scale both columns to 100%.
  • Find the length of the school day, and check that your answer is larger than 75 minutes.

Solve: $42 is 60% of the original price. Complete the ratio table to reach 100%.

PercentDollars
60%42
10%
100%

Stuck? Open one hint at a time.

Hint 1

60% ÷ 6 = 10%, so divide the dollars by 6 as well.

Hint 2

42 ÷ 6 = 7, so 10% is $7.

Hint 3

100% is ten of those steps: 7 × 10 = ?

Check my answer
✅ Answers:
  • 10% → Dollars: 7
  • 100% → Dollars: 70

Turn & TalkSay it out loud with a partner

Akela paid $42 for a sweater, and that was 60% of the original price. BEFORE computing: will the original price be more or less than $42? How do you know?

Try starting with:

  • The original price will be ___ than $42 because $42 is only ___ of it.El precio original será ___ que $42 porque $42 es solo ___ de él.
  • The part is ___, the percent is ___, and the whole is ___.La parte es ___, el porcentaje es ___ y el entero es ___.
Use these wordspartwholepercent