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RATIO COMPARISONS - Test of English as Foreign Language (TOEFL)

(A) Calculation methods for Ratio comparisons:

There could be four broad cases when you might be required to do ratio comparisons: The table below clearly illustrates these:

In cases 2 and 4 in the table, calculations will be necessitated. In such a situation, the following processes can be used for ratio comparisons.


1. The Cross Multiplication Method

Two ratios can be compared using the cross multiplication method as follows. Suppose you have to compare 12/ 15 with 15/ 19

Then, to test which ratio is higher cross multiply and compare 12 19 and 15 17.

If 12 19 is bigger the Ratio 12/ 17 will be bigger. If 15 17 is higher, the ratio 15/ 19 will be higher.


In this case, 15 17 being higher, the Ratio 15/ 19 is higher.

Thus, this method will not be able to tell you the answer if you have to compare with

3743/5624 with 3821/5783.


2. Percentage Value Comparison Method


Suppose you have to compare: 173/212 with 181/241

In such a case just by estimating the 10% ranges for each ratio you can clearly see that —

the first ratio is > 80% while the second ratio is < 80%

Hence, the first ratio is obviously greater.


This method is extremely convenient if the two ratios have their values in different 10% ranges.


However, this problem will become slightly more difficult, if the two ratios fall in the same 10% range. Thus, if you had to compare 173/ 212 with 181/ 225, both the values would give values between 80 – 90%. The next step would be to calculate the 1% range.


The first ratio here is 81 – 82% while the second ratio lies between 80 – 81% Hence the first ratio is the larger of the two.


Note: For this method to be effective for you, you will first need to master the percentage rule method for calculating the percentage value of a ratio. Hence if you cannot see that 169.6 is 80% of 212 or for that matter that 81% of 212 is 171.72 and 82% is 173.84 you will not be able to use this method effectively. (This is also true for the next method.) However, once you can calculate percentage values of 3 digit ratios to 1% range, there is not much that can stop you in comparing ratios. The CAT, TOEFL and all other aptitude exams normally do not challenge you to calculate further than the 1% range when you are looking at ratio comparisons.


3. Numerator Denominator Percentage Change Method


There is another way in which you can compare close ratios like 173/ 212 and 181/ 225. For this method, you need to calculate the percentage changes in the numerator and the denominator.


173 181 is a % increase of 4 – 5%

While 212 225 is a % increase of 6 – 7%.

In this case, since the denominator is increasing more than the numerator, the second ratio is smaller. This method is the most powerful method for comparing close ratios― provided you are good with your percentage rule calculations.


(B) Method for calculating the value of a percentage change in the ratio:


PCG (Percentage Change Graphic) gives us a convenient method to calculate the value of the percentage change in a ratio. Suppose, you have to calculate the percentage change between 2 ratios. This has to be done in two stages as:

Thus if 20/ 40 becomes 22/ 50

Effect of numerator = 20 22( 10% increase)

Effect of denominator = 50 40( 25% decrease) (reverse fashion) Overall effect on the ratio:

Hence, overall effect = 17.5% decrease.