Adding And Subtracting Fractions: A Step-by-Step Guide

Hey guys! Let's dive into the world of fractions and learn how to add and subtract them like pros. It might seem tricky at first, but with a little practice, you'll be breezing through these problems in no time. Our main goal here is to find a common denominator, which is the key to unlocking these calculations. So, let's get started!

a) ${\frac{2}{15} + \frac{1}{16} =}$

Okay, so our first problem is adding ${\frac{2}{15}}$ and ${\frac{1}{16}}$. The first thing we need to do is find the least common multiple (LCM) of the denominators, which are 15 and 16. This LCM will be our common denominator. To find the LCM, we can list the multiples of each number or use prime factorization.

Let's use prime factorization. First, we break down 15 and 16 into their prime factors:

• 15 = 3 x 5
• 16 = 2 x 2 x 2 x 2 = ${2^4}$

Now, to find the LCM, we take the highest power of each prime factor that appears in either factorization. That means we need 2 (to the power of 4), 3, and 5.

LCM (15, 16) = ${2^4}$ x 3 x 5 = 16 x 3 x 5 = 240

Great! Our common denominator is 240. Now we need to convert both fractions to have this denominator. To do that, we multiply both the numerator and the denominator of each fraction by the number that makes the denominator equal to 240.

For ${\frac{2}{15}}$, we need to multiply the denominator 15 by 16 to get 240. So, we also multiply the numerator 2 by 16:

${\frac{2}{15} = \frac{2 \times 16}{15 \times 16} = \frac{32}{240}}$

For ${\frac{1}{16}}$, we need to multiply the denominator 16 by 15 to get 240. So, we also multiply the numerator 1 by 15:

${\frac{1}{16} = \frac{1 \times 15}{16 \times 15} = \frac{15}{240}}$

Now we can add the fractions:

${\frac{32}{240} + \frac{15}{240} = \frac{32 + 15}{240} = \frac{47}{240}}$

So, ${\frac{2}{15} + \frac{1}{16} = \frac{47}{240}}$. And that's our answer for the first one! Remember, the key is finding that LCM and converting the fractions correctly.

b) ${\frac{2}{65} + \frac{7}{78} =}$

Next up, we have ${\frac{2}{65} + \frac{7}{78}}$. Again, our mission is to find the least common multiple (LCM) of 65 and 78. Let's use prime factorization to make it easier.

First, we break down 65 and 78 into their prime factors:

• 65 = 5 x 13
• 78 = 2 x 3 x 13

To find the LCM, we take the highest power of each prime factor that appears in either factorization. That means we need 2, 3, 5, and 13.

LCM (65, 78) = 2 x 3 x 5 x 13 = 390

Alright, our common denominator is 390. Now, let's convert both fractions to have this denominator.

For ${\frac{2}{65}}$, we need to multiply the denominator 65 by 6 to get 390. So, we also multiply the numerator 2 by 6:

${\frac{2}{65} = \frac{2 \times 6}{65 \times 6} = \frac{12}{390}}$

For ${\frac{7}{78}}$, we need to multiply the denominator 78 by 5 to get 390. So, we also multiply the numerator 7 by 5:

${\frac{7}{78} = \frac{7 \times 5}{78 \times 5} = \frac{35}{390}}$

Now we can add the fractions:

${\frac{12}{390} + \frac{35}{390} = \frac{12 + 35}{390} = \frac{47}{390}}$

So, ${\frac{2}{65} + \frac{7}{78} = \frac{47}{390}}$. Fantastic! We're on a roll.

c) ${\frac{7}{24} - \frac{5}{36} =}$

Now let's tackle some subtraction! We have ${\frac{7}{24} - \frac{5}{36}}$. The process is very similar to addition; we still need to find a common denominator. So, let's find the LCM of 24 and 36.

First, we break down 24 and 36 into their prime factors:

• 24 = 2 x 2 x 2 x 3 = ${2^3}$ x 3
• 36 = 2 x 2 x 3 x 3 = ${2^2}$ x ${3^2}$

To find the LCM, we take the highest power of each prime factor that appears in either factorization. That means we need ${2^3}$ and ${3^2}$.

LCM (24, 36) = ${2^3}$ x ${3^2}$ = 8 x 9 = 72

Great, our common denominator is 72. Now we need to convert both fractions to have this denominator.

For ${\frac{7}{24}}$, we need to multiply the denominator 24 by 3 to get 72. So, we also multiply the numerator 7 by 3:

${\frac{7}{24} = \frac{7 \times 3}{24 \times 3} = \frac{21}{72}}$

For ${\frac{5}{36}}$, we need to multiply the denominator 36 by 2 to get 72. So, we also multiply the numerator 5 by 2:

${\frac{5}{36} = \frac{5 \times 2}{36 \times 2} = \frac{10}{72}}$

Now we can subtract the fractions:

${\frac{21}{72} - \frac{10}{72} = \frac{21 - 10}{72} = \frac{11}{72}}$

So, ${\frac{7}{24} - \frac{5}{36} = \frac{11}{72}}$. Subtraction is just like addition, but in reverse! Keep an eye on those numerators.

d) ${\frac{5}{14} + \frac{3}{28} + \frac{1}{42} =}$

Alright, let's level up and add three fractions together: ${\frac{5}{14} + \frac{3}{28} + \frac{1}{42}}$. The principle is the same: we need a common denominator for all three fractions. Let's find the LCM of 14, 28, and 42.

First, we break down 14, 28, and 42 into their prime factors:

• 14 = 2 x 7
• 28 = 2 x 2 x 7 = ${2^2}$ x 7
• 42 = 2 x 3 x 7

To find the LCM, we take the highest power of each prime factor that appears in any of the factorizations. That means we need ${2^2}$, 3, and 7.

LCM (14, 28, 42) = ${2^2}$ x 3 x 7 = 4 x 3 x 7 = 84

Our common denominator is 84. Now, let's convert all three fractions to have this denominator.

For ${\frac{5}{14}}$, we need to multiply the denominator 14 by 6 to get 84. So, we also multiply the numerator 5 by 6:

${\frac{5}{14} = \frac{5 \times 6}{14 \times 6} = \frac{30}{84}}$

For ${\frac{3}{28}}$, we need to multiply the denominator 28 by 3 to get 84. So, we also multiply the numerator 3 by 3:

${\frac{3}{28} = \frac{3 \times 3}{28 \times 3} = \frac{9}{84}}$

For ${\frac{1}{42}}$, we need to multiply the denominator 42 by 2 to get 84. So, we also multiply the numerator 1 by 2:

${\frac{1}{42} = \frac{1 \times 2}{42 \times 2} = \frac{2}{84}}$

Now we can add the fractions:

${\frac{30}{84} + \frac{9}{84} + \frac{2}{84} = \frac{30 + 9 + 2}{84} = \frac{41}{84}}$

So, ${\frac{5}{14} + \frac{3}{28} + \frac{1}{42} = \frac{41}{84}}$. Adding more fractions just means more converting, but the method stays the same.

e) ${\frac{7}{36} + ...}$

And that's it for now, folks! You've learned how to add and subtract fractions by finding a common denominator. Keep practicing, and you'll become a fraction master in no time! Remember, the LCM is your friend, and double-check your multiplication when converting fractions. Good luck, and have fun with fractions!