What Is 60 of 75 of 60? Understanding Percent‑of‑Percent Calculations
When you encounter a phrase like “60 of 75 of 60”, the most common interpretation in everyday math is that you are being asked to find 60 % of 75 % of 60. Now, this type of problem appears frequently in schoolwork, finance, statistics, and real‑life situations such as calculating discounts, tax rates, or population subsets. Below we break down the concept step‑by‑step, explain why the calculation works, provide practical examples, and answer frequently asked questions so you can confidently solve similar problems on your own Practical, not theoretical..
1. Breaking Down the Phrase
| Part of the phrase | Mathematical meaning | How to write it |
|---|---|---|
| 60 (first) | 60 % (or 0.60 | |
| 75 (middle) | 75 % (or 0.Think about it: 60 as a decimal) | 60 % → 0. 75 as a decimal) |
Thus “60 of 75 of 60” translates to:
[ 0.60 \times \bigl(0.75 \times 60\bigr) ]
The parentheses indicate that we first find 75 % of 60, then take 60 % of that result Most people skip this — try not to..
2. Step‑by‑Step Calculation
-
Find 75 % of 60
[ 0.75 \times 60 = 45 ] -
Find 60 % of the result (45)
[ 0.60 \times 45 = 27 ]
Answer: 60 of 75 of 60 = 27.
You can also combine the two percentages into a single multiplier:
[ 0.Even so, 60 \times 0. 75 = 0.45 \quad\text{(45 %)} \ 0.
So the problem is equivalent to asking for 45 % of 60, which yields the same result Easy to understand, harder to ignore..
3. Why the Order Does Not Matter (Commutative Property)
Multiplication is commutative and associative, meaning you can rearrange the factors without changing the product:
[ 0.75 \times 0.So 60 \times 0. 75 \times 60 = 0.60 \times 60 = 60 \times 0.60 \times 0.
If you mistakenly interpreted the phrase as “60 % of (75 % of 60)” versus “75 % of (60 % of 60)”, you would still get 27 because the two percentages multiply to the same combined factor (0.45). This property is useful when solving more complex chains of percentages.
4. Real‑World Applications
a. Successive Discounts
A store offers a 25 % discount on an item, then an additional 40 % discount on the reduced price. To find the final price of a $60 item:
- First discount: 25 % off → pay 75 % of $60 = $45.
- Second discount: 40 % off → pay 60 % of $45 = $27.
The final price is $27, exactly the same as our original calculation (just with different percentages) That alone is useful..
b. Population Subsets
Suppose a town has 60,000 residents. 75 % of them are adults, and 60 % of those adults are employed. Number of employed adults:
[ 0.75 \times 60{,}000 = 45{,}000 \quad\text{(adults)} \ 0.60 \times 45{,}000 = 27{,}000 \quad\text{(employed adults)} ]
c. Compound Interest Approximation
For very short periods, interest can be approximated by successive percentage increases. If an investment grows 75 % in the first period and then 60 % in the second, the overall growth factor is 0.75 × 0.60 = 0.45, meaning a 45 % increase over the original amount.
5. Common Mistakes and How to Avoid Them
| Mistake | Why It Happens | Correct Approach |
|---|---|---|
| Adding percentages (e. | ||
| Misplacing the decimal (e.Think about it: g. , using 60 instead of 0.But , 60 % + 75 % = 135 %) | Confusing “of” with “and” | Remember “of” means multiplication, not addition. g., 60 % of 60 after finding 75 % of 60) |
| Applying the second percentage to the original number (e. 60) | Overlooking the conversion from percent to decimal | Convert each percent to a decimal by dividing by 100 before multiplying. Here's the thing — g. |
| Ignoring order of operations | Assuming the phrase can be read left‑to‑right without parentheses | Use parentheses to clarify which operation happens first, or rely on the commutative property if you’re comfortable. |
A quick sanity check: the final answer must be smaller than the original number when you are taking percentages less than 100 % successively. Since both 60 % and 75 % are below 100 %, the result (27) is indeed less than 60.
6. Practice Problems
Try solving these on your own before checking the answers.
- What is 30 % of 50 % of 200?
- Calculate 80 % of 90 % of 150.
- A jacket costs $120. It receives a 20 % discount, then a further 15 % discount. What is the final price?
Answers:
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