# Largest Fraction

Which fraction is the largest?

1. 5/8
2. 7/12
3. 11/16
4. 13/24
5. 31/ 48

This question was given in the lower primary level (9 years and younger) of the 3rd International Vedic Mathematics Olympiad (IVMO 2023) last November 25 2023. Perhaps, due to time pressure, many competitors did not get the correct answer.

In chapter 11 of our book, Algebra Made Easy as Arithmetic, we discussed a method of comparing two fractions. After cross-multiplication(multiplying the numerator of one fraction by the denominator of the other), the fraction with the numerator giving the bigger product is bigger.

For example, when comparing 3/4 and 6/7, cross-multiplication will give 3 x 7 = 21 and 6 x 4 = 24. The bigger product is 24 which was obtained using 6, the numerator of 6/7. Therefore, 6/7 is the bigger of the two fractions.

Cross-multiplication works because it is a simplified way of making the fractions similar or having the same denominators. 3/4 can be multiplied by 7/7 to give 21/28 while 6/7 can be multiplied by 4/4 to give 24/28. Now it is easy to see that with the two fractions having the same denominator, the one having the bigger numerator is bigger: 24/28 > 21/28.

In the problem at hand, we have to find the biggest of 5 fractions. But just by looking at the denominators, we can see that the biggest is 48 and the rest are factors of 48.  We can just convert all of them into similar fractions with 48 as denominators.

We can solve this mentally using the following steps:

1. Start with choice E) 31/48; 31 is so far the biggest numerator.
2. In choice D, 13/24, we can see that 48 is twice 24, so we have to multiply 13 also by 2 to get 26. 31 is still bigger.
3. We have 11/16 for choice C. We see that 16 is a third of 48, so we have to multiply 11 by 3 to get 33 which is bigger than 31.
4. We need to multiply both terms of option B) 7/12 by 4 to have a denominator of 4. 4 x 7 gives 28, which is less than 33
5. The first choice is A) 5/8. Multiplying it by 6/6 will result into 30/48.

Choice C) 11/16 or 33/48 is the biggest among the 5 fractions.