To add these two fractions together, 3/7 + 5/12, we need a common denominator. The common denominator is simply the least common multiple for the 2 denominators. So here, our first step is to find the least common multiple for 7 and 12.
7 doesn't factor but 12 factors as 2 x 6 and 6 factors as 2 x 3.
When finding the least common multiple, we're looking for factors that match up and here we have no factors that match up so our least common multiple or <em>lcm</em> is 7 · 12 or 84. So our least common denominator for these two fractions is 84.
In order to get a denominator of 84 in the first fraction, we must multiply our denominator of 7 by 12. The problem is we can't just multiply the denominator of a fraction by 12 because that changes its value.
However, if we also multiply the numerator of the fraction by 12, we're actually multiplying the fraction by 12/12 which is 1 and when we multiply anything by 1, it doesn't change its value.
So our first fraction becomes 36/84. Notice that 35/84 has the same value as 3/7.
For our second fraction we have a 12 in the denominator and we want a common denominator of 84 so we must multiply this 12 by 7 and when we multiply the denominator of a fraction by 7, we must also multiply the numerator by 7. When we do this, we get the fraction 35/84.
Now adding across the numerators, we get 71/84.
My explanation for why this fraction is less than 1 is provided in the image that is attached below.
To add unlike fractions such as 3/8 + 4/6, first find a common denominator. The common denominator will be the least common multiple for 8 and 6 which is 24.
To get a 24 in the denominator of 3/8, we multiply top and bottom by 3 to get 9/24 and we multiply top and bottom of 4/6 by 4 to get 16/24.
So we have 9/24 + 16/24 which is 25/24 or 1 and 1/4.
To write an improper fraction as a mixed number, simply divide the denominator into the numerator.
Ask me in the comments if you need help understanding that.
My work for these 2 problems is attached below.
