# Miscellaneous

## Percent of a Percent

x% of y = $\frac{x}{100}$ × y

z% of x% of y = $\frac{z}{100}$ × $\frac{x}{100}$ × y

and so on.

You will not that the order in which we apply these percentages is not important:

z% of x% of y = $\frac{z}{100}$ × $\frac{x}{100}$ × y

x% of z% of y = $\frac{x}{100}$ × $\frac{z}{100}$ × y

So, z% of x% of y = x% of z% of y

##### Q. What is 20% of 50% of 90?

Explanations :

Explanation 1:

50% of 90 = $\frac{50}{100}$ × 90 = 45

20% of 45 = $\frac{20}{100}$ × 45 = 9

Explanation 2: Using the concept of Multiplying Factors

20% of 50% of 90 = 0.2 × 0.5 × 90 = 9

## A percentage increase is how many times the original value?

If the percentage increase is 100%, then the new value of something is 2 times the previous given value.

If the percentage increase is 200%, then the new value of something is 3 times the previous given value.

If the percentage increase is 50%, then the new value of something is 1.5 times the previous given value.

So, we can generalize:

If the percentage increase is p%, then the new value of something is (1 + $\frac{p}{100}$) times the old value.

Or we can say that:

If the new value of something is p times the old value, then the percentage increase is (p - 1)100%.

##### Q. If salary of Mragank is increases by 150%, then his new salary is how many times his earlier salary?

Explanation:

If the percentage increase is p%, then the new value of something is (1 + $\frac{p}{100}$) times the old value.

So, New salary = (1 + $\frac{p}{100}$) times the earlier salary = (1 + $\frac{150}{100}$) times the earlier salary = 2.5 times the earlier salary

##### Q. If X = 5.25 Y, then by how much percent is X more than Y?

Explanation:

X is 5.25 times the value of Y.

So, Percentage increase from Y to X = (p - 1)100% = (5.25 -1)100 = 425%

## p times Vs. p times more

Compare these two:

• If the new value of something is p times the previous given value, then the percentage increase is (p - 1)100%.

• If the new value of something is p times more than the previous given value, then the percentage increase is 100p%.

##### Q. If price of onion increases to 3 times of its previous value, then what is the percent increase?

Explanation:

If initial price was Rs 100, new price = Rs 300

If the new value of something is p times the previous given value, then the percentage increase is (p - 1)100%.

So, Percentage increase = (p - 1)100% = (3 - 1)100 = 200%

##### Q. If price of onion becomes 3 times more than its previous value, then what is the percent increase?

Explanation:

If initial price was Rs 100, new price = 100 + 300 = Rs 400

If the new value of something is p times more than the previous given value, then the percentage increase is 100p%.

So, Percentage increase = 100p% = 300%

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