# Largest Fraction

**Which fraction is the largest?**

**5/8****7/12****11/16****13/24****31/ 48**

This question was given in the lower primary level (9 years and younger) of the 3^{rd} 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:

- Start with choice
**E) 31/48**;**31**is so far the biggest numerator. - 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. - 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.** - 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** - 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.