Here is a problem posed by the ancient Indian Mathematician Mahaviracharya (850 C.E.). Add together , , , and . What do you get? How can we find this sum efficiently?

Finding Common Ground | FIO

Question 22

Here is a problem posed by the ancient Indian Mathematician Mahaviracharya (850 C.E.). Add together $\frac{8}{15}$ , $\frac{1}{20}$ , $\frac{7}{36}$ , $\frac{11}{63}$ and $\frac{1}{21}$ . What do you get? How can we find this sum efficiently?

Solution

To add fractions, we must first find a common bottom number.

Step 1 — Find the Least Common Multiple (LCM)

First, let us list all the denominators. The denominators are 15, 20, 36, 63, and 21. We find the prime factors for each number. This helps us find the smallest common multiple.

$15 = 3 \times 5$

$20 = 2 \times 2 \times 5 = 2^2 \times 5$

$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

$63 = 3 \times 3 \times 7 = 3^2 \times 7$

$21 = 3 \times 7$

Now, we take the highest power of each prime factor. The highest power of 2 is $2^2$ . The highest power of 3 is $3^2$ . The highest power of 5 is $5^1$ . The highest power of 7 is $7^1$ . We multiply these highest powers together. This gives us the Least Common Multiple.

$\text{LCM}(15, 20, 36, 63, 21) = 2^2 \times 3^2 \times 5 \times 7$

$= 4 \times 9 \times 5 \times 7$

$= 36 \times 35$

$\boxed{1260}$

Diagram 1

Step 2 — Convert fractions to the common denominator

We will change each fraction to have a denominator of 1260. For each fraction, we divide 1260 by its original denominator. Then we multiply both the top and bottom by that number.

For $\frac{8}{15}$ : We divide 1260 by 15, which is 84. We multiply 8 and 15 by 84.

$\frac{8}{15} = \frac{8 \times 84}{15 \times 84} = \frac{672}{1260}$

For $\frac{1}{20}$ : We divide 1260 by 20, which is 63. We multiply 1 and 20 by 63.

$\frac{1}{20} = \frac{1 \times 63}{20 \times 63} = \frac{63}{1260}$

For $\frac{7}{36}$ : We divide 1260 by 36, which is 35. We multiply 7 and 36 by 35.

$\frac{7}{36} = \frac{7 \times 35}{36 \times 35} = \frac{245}{1260}$

For $\frac{11}{63}$ : We divide 1260 by 63, which is 20. We multiply 11 and 63 by 20.

$\frac{11}{63} = \frac{11 \times 20}{63 \times 20} = \frac{220}{1260}$

For $\frac{1}{21}$ : We divide 1260 by 21, which is 60. We multiply 1 and 21 by 60.

$\frac{1}{21} = \frac{1 \times 60}{21 \times 60} = \frac{60}{1260}$

Step 3 — Add the numerators

Now all fractions have the same denominator. We can add their top numbers (numerators). The denominator stays the same.

$\text{Sum} = \frac{672}{1260} + \frac{63}{1260} + \frac{245}{1260} + \frac{220}{1260} + \frac{60}{1260}$

$= \frac{672 + 63 + 245 + 220 + 60}{1260}$

$= \frac{1260}{1260}$

$= 1$

The sum of all the fractions is 1.

Answer

(i) The sum of the given fractions is 1. (ii) We find the LCM of the denominators to add them efficiently.

More questions in FIO

List all the factors of the following numbers:

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(b) 105

(d) 360 (this number has 24 factors)

(e) 840 (this number has 32 factors)

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(d) 370, 592

(e) 81, 243

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Share your observations with the class.

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(b) Make a general statement about such numbers. Describe such number pairs using algebra.

Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold.

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