## Comparing fractions and ordering them – An easy approach

The ability to use interchangeably, ratios, fractions and decimal numbers will be very useful in QA section of the competitive examinations. A ratio is a numerical presentation comparing two similar things measured in the same unit of measurement. In its basic form, the two components are relative primes. That is they cannot be further simplified through division by a common factor other than 1. The same ratio can be expressed as a fraction with the first component of ratio in the numerator and second component in the denominator.

For example the ratio between two quantities, 2:3 is equivalent to the fraction 2/3 . While ratio simply gives an idea of relative values of the two quantities, fraction 2/3 is an expression,  which gives the value of the first quantity as a proportion of the second component of the ratio. When the denominator of the fraction is raised to 100 by multiplying both numerator and denominator by an appropriate number we get a single no 66.66. So, all the three forms can be used equivalently depending on the context of the problems.

Often we have to handle problems involving comparison of a set of fractions and finding out the highest, smallest, second highest etc. In other words, we have to find out the increasing or decreasing order of the fractions.  When the numerators are equal, it is simple. The smaller the denominator the higher is the value of the fraction and vice versa.

When the denominators of the fractions are not equal, we can use the following three methods

### LCM approach

In this method we make the denominators equal using LCM of the denominators. We can find the LCM of all the denominators and multiply all the fractions with appropriate numbers to make all the denominators equal to LCM. We will get different numerators but with all denominators equal to LCM. we can order the fractions.

### Example:

Which is bigger?  3/7 or 4/9. Taking LCM of 7 and 9 we get 63

To make denominator of first fraction as 63 we multiply both numerator and denominator by 9 (which is LCM divided by denominator) and the fraction becomes 27/63. Similarly second fraction will be equal to 28/63.  Now it is easy to interpret that second fraction 4/9 is the bigger one.

When we have to compare only two fractions we need not even look for LCM of the denominators. We can take the product of the two denominators as common denominator. Then we can multiply numerator of first fraction by denominator of the second fraction and similarly numerator of the second fraction by denominator of the first fraction and compare. In the above case we compare 3 × 9 and 4 × 7 and conclude  that the second fraction is bigger. When the denominators of the fractions are relative prime numbers, their product is equal to LCM.

When there are more than 2 fractions LCM would be better. When we have more than 3 fractions with different numerators and denominators the calculation becomes more time consuming, especially when the denominators are relatively prime to each other like say 3, 7 and 8.

### 2. Conversion of fractions to decimal numbers

In this approach we convert  all fractions into decimal numbers and then arrange them in order. This approach can also be time consuming. Consider the following example.

Arrange the following fractions in ascending order

4/17, 5/21, 7/23 Under LCM method, we have to evaluate 4 × 21 × 23, 5 × 17 ×23, and 7 ×17 × 23 which would be time consuming.  So we can convert the numbers into decimals as follows;

4/17  =  400/17 = 23.53

5/21  =  500/21 = 23.81

7/23  =  700/23  = 30.43

So the fractions in ascending order are  4/17, 5/21, 7/23. Again this is also time consuming as division by prime numbers are involved. Now we consider a simpler approach.

### 3. Truncated Decimal conversion approach

The approach is very simple. Multiply the numerators of all the fractions by multiples of 10 in stages. If the denominators are all whole numbers we multiply the numerators by 10 and divide by respective denominators. If the denominators contain numbers with decimals, for each significant digit after the decimal point in the denominator we add one zero to the multiplier of the numerators. For example if there are two denominators , say 11.4 and 13.67 we multiply all the numerators by 100 and then find out the dividends.

Find out the dividends and leave the remainders. If the dividends can be arranged in ascending order we get the order.

Example:  Arrange in descending order   4/13, 7/17, 11/21,  14/23

Since the denominators are all whole nos multiply the numerators by 10 and find the dividends. This we can do mentally.

We get       40/13, 70/17, 110/21,  140/23

Values are 3 + fraction, 4 + fraction, 5 + fraction, 6+ fraction

Since fractions are all less than 1 we get the order of the fractions from dividends

Suppose two are three dividends are equal we multiply only those fractions by 100 and evaluate

Consider the example 4/17, 6/23, 10/27

Multiplying by 10 and finding the dividends we get  2+, 2+, 3+ (third is biggest)

Multiplying the first and last fractions by 10 again (totally 100) we get 400/17 and 600/23

We  get  23+, 26+ (second is bigger)

Thus we get the order as   4/17, 5/21, 7/23

We can just see how simple these  calculations are. For n fractions to be compared the maximum no of steps required  is n-1.

This approach is very handy in DI problems. We invariably get a question involving comparison of a no of percentage values, usually 4 or more , like for instance in which year the company’s profit percentage is highest. We will have to calculate 4 or more percentages involving different numerators and different denominators. Instead of calculating the percentages straightaway,  we can take the fractions and adopt the above approach. In one or two steps, we will get the answer.