Percents and Ratios Flashcards
Percent
Fundamentally, a percent is a fraction. The percent sign can be thought of as a stylized version of “divided by 100”
x% = x/100
Converting percents to decimals
Done by simply dividing by 100, so we move the decimal point two places to the left.
Ex: 42.5% = 0.425
Ex: 0.25% = 0.0025
Converting decimals to percents
Accomplished by multiplying the decimal by 100, so we move the decimal point two places to the right.
Ex: 0.68 = 68%
Ex: 2.3 = 230%
Converting percents to fractions
Simply put the percent over 100 - you may have to simplify the fraction from there.
Ex: 20% = 20/100 = 2/10 = 1/5
Ex: 0.02% = 0.02/100 = 2/10,000 = 1/5,000
Converting fractions to percents
First, you must know the fraction to decimal conversion (like 1/2 = 0.5)
Once you have the decimal, you convert to a percent by multiplying by 100, and therefore moving the decimal point two places to the right.
Ex: 3/8 = 0.375 = 37.5%
If the fraction has 100 (or other power of 10) in the denominator, however, it’s very easy to change to a decimal, which would give us the percent.
Ex: 17/1000 = 1.7/100 = 0.017 = 1.7%
Approximating Fractions as Percents
The GMAT often asks us to approximate percents from fractions or division.
Ex: 8/33
To get to a number we know the decimal conversion for, we can multiply both the numerator and denominator by 3.
8/33 x 3/3 = 24/99
24/99 is just slightly greater than 24/100 (because the numerators are the same but the denominator is smaller).
Therefore, 8/33 is slightly more than 24%
Number Sense and Percents
A more efficient way to calculate percents that involves finding 10% and sometimes 1% of the number.
Ex: what is 37% of 700?
- we know 10% of 700 = 70
- we know 1% of 700 = 7
10% + 10% + 10% + 7(1%) = 37% SO
3(70) + 7(7) =
210 + 49 =
259
Percent Increases
Could be phrased as:
- y increased by 30%
- x is 30% greater than y
Think about it this way: if x increases by 30%, the whole part of x is still there PLUS 30%
In general, if a problem asks about a p% increase, the multiplier = (1 + p% as a decimal)
The multiplier for a 46% increase = (1 + 0.46)
= 1.46
Ex: after a 30% increase, the price of an item is $78. What was the original price?
78 = 1.3x
x = 78/1.3 = 780/13
= $60
Percent Decreases
Could be phrased as:
- y decreased by 30%
- x is 30% less than y
Let’s think about this: if y decreases by 30%, most of the whole original part of y is still there except for the 30% now missing.
Therefore, the multiplier for a 30% decrease is:
1 - 0.30 = 0.70
In general, if a problem talks about a p% decrease, the multiplier = 1 - (p% as a decimal)
Ex: After an item was discounted 80% the new price is 150. What was the original price?
150 = (1 - 0.80)x 150 = 0.2x x = 150/0.2 = 1500/2 = 750.
Finding the percent in a word problem
Some problems will give you the starting abs ending values and ask you to find the percent increase or decrease.
Since (new) = (multiplier)(old)
Multiplier = (new)/(old)
Don’t forget we have to then change the multiplier back into a percent (1 - or 1+ depending on if it’s an increase or decrease!)
Ex: the price of an item increased from 200 to 800. What was the percent increase?
M = 800/2 M = 4 (which is 1 + 3 since this was a percent increase)
The correct answer is a 300% increase.
Sequential percent changes
For a series of percent changes, multiply the individual multipliers together to calculate overall percent change.
Ex: An item cost 100, increased by 30%, then an employee bought it at a 30% discount. What did the employee pay?
X = (100)(1.3)(0.7)
X = (130)(0.7)
= $91.0
Ex: price of stock increased 20%, decreased 50%, then increased 40%. What is the percent change?
m = (1.2)(0.5)(1.4)
= (12)(5)(14)
= 840 (but 3 decimal points to the left!)
= .84
Because the multiplier is 0.84, that means the percent change equals a decrease of 16% overall.
% increase or % decrease
% increase/decrease =
change (subtract your find the difference)/initial amount
Sequential percent changes
For a series of percent changes, multiply the individual multipliers together to calculate overall percent change.
Ex: An item cost 100, increased by 30%, then an employee bought it at a 30% discount. What did the employee pay?
X = (100)(1.3)(0.7)
X = (130)(0.7)
= $91.0
Ex: price of stock increased 20%, decreased 50%, then increased 40%. What is the percent change?
m = (1.2)(0.5)(1.4)
= (12)(5)(14)
= 840 (but 3 decimal points to the left!)
= .84
Because the multiplier is 0.84, that means the percent change equals a decrease of 16% overall.
