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WHAT IS ADDING & SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS?

Two fractions have unlike denominators if the numbers in the denominators are not the same. You can use multiplication and division to rewrite fractions to have a common denominator.

To better understand adding and subtracting fractions with unlike denominators…

WHAT IS ADDING & SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS?. Two fractions have unlike denominators if the numbers in the denominators are not the same. You can use multiplication and division to rewrite fractions to have a common denominator. To better understand adding and subtracting fractions with unlike denominators…

## LET’S BREAK IT DOWN!

### Algorithm for Finding a Common Denominator

Let's say you want to add [ggfrac]1/3[/ggfrac]+ [ggfrac]2/5[/ggfrac]. You know that you need to find a common denominator. Starting with [ggfrac]1/3[/ggfrac], list equivalent fractions: [ggfrac]1/3[/ggfrac] × [ggfrac]2/2[/ggfrac] = [ggfrac]2/6[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]3/3[/ggfrac]= [ggfrac]3/9[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]4/4[/ggfrac] = [ggfrac]4/12[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]5/5[/ggfrac] = [ggfrac]5/15[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]6/6[/ggfrac] = [ggfrac]6/18[/ggfrac]. None of those fractions have a denominator of 5, so you need to find equivalent fractions for [ggfrac]2/5[/ggfrac] as well: [ggfrac]2/5[/ggfrac] × [ggfrac]2/2[/ggfrac]= [ggfrac]4/10[/ggfrac], [ggfrac]2/5[/ggfrac] × [ggfrac]3/3[/ggfrac]= [ggfrac]6/15[/ggfrac], [ggfrac]2/5[/ggfrac] × [ggfrac]4/4[/ggfrac]= [ggfrac]8/20[/ggfrac]. Both [ggfrac]1/3[/ggfrac] and [ggfrac]2/5[/ggfrac] have an equivalent fraction with a denominator of 15. Rewrite the original addition as [ggfrac]5/15[/ggfrac]+ [ggfrac]6/15[/ggfrac]= [ggfrac]11/15[/ggfrac].

There is a faster way to find a common denominator. If you multiply the numerator and the denominator of each fraction by the denominator of the other fraction, you always get fractions with a common denominator. Here's how this works: Starting with [ggfrac]1/3[/ggfrac], you see that the other fraction, [ggfrac]2/5[/ggfrac], has a denominator of 5 so you multiply the numerator and denominator by 5: [ggfrac]1/3[/ggfrac] × [ggfrac]5/5[/ggfrac] = [ggfrac]5/15[/ggfrac]. Next, you see that [ggfrac]1/3[/ggfrac] has a denominator of 3, so you multiply the numerator and denominator of the other fraction by 3: [ggfrac]2/5[/ggfrac] × [ggfrac]3/3[/ggfrac]= [ggfrac]6/15[/ggfrac]. Rewrite the original addition as [ggfrac]5/15[/ggfrac]+ [ggfrac]6/15[/ggfrac]= [ggfrac]11/15[/ggfrac]. Try this one yourself. Add [ggfrac]1/6[/ggfrac] + [ggfrac]3/4[/ggfrac] by finding a common denominator using the faster way.

Algorithm for Finding a Common Denominator Let's say you want to add [ggfrac]1/3[/ggfrac]+ [ggfrac]2/5[/ggfrac]. You know that you need to find a common denominator. Starting with [ggfrac]1/3[/ggfrac], list equivalent fractions: [ggfrac]1/3[/ggfrac] × [ggfrac]2/2[/ggfrac] = [ggfrac]2/6[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]3/3[/ggfrac]= [ggfrac]3/9[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]4/4[/ggfrac] = [ggfrac]4/12[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]5/5[/ggfrac] = [ggfrac]5/15[/ggfrac], [ggfrac]1/3[/ggfrac] × [ggfrac]6/6[/ggfrac] = [ggfrac]6/18[/ggfrac]. None of those fractions have a denominator of 5, so you need to find equivalent fractions for [ggfrac]2/5[/ggfrac] as well: [ggfrac]2/5[/ggfrac] × [ggfrac]2/2[/ggfrac]= [ggfrac]4/10[/ggfrac], [ggfrac]2/5[/ggfrac] × [ggfrac]3/3[/ggfrac]= [ggfrac]6/15[/ggfrac], [ggfrac]2/5[/ggfrac] × [ggfrac]4/4[/ggfrac]= [ggfrac]8/20[/ggfrac]. Both [ggfrac]1/3[/ggfrac] and [ggfrac]2/5[/ggfrac] have an equivalent fraction with a denominator of 15. Rewrite the original addition as [ggfrac]5/15[/ggfrac]+ [ggfrac]6/15[/ggfrac]= [ggfrac]11/15[/ggfrac]. There is a faster way to find a common denominator. If you multiply the numerator and the denominator of each fraction by the denominator of the other fraction, you always get fractions with a common denominator. Here's how this works: Starting with [ggfrac]1/3[/ggfrac], you see that the other fraction, [ggfrac]2/5[/ggfrac], has a denominator of 5 so you multiply the numerator and denominator by 5: [ggfrac]1/3[/ggfrac] × [ggfrac]5/5[/ggfrac] = [ggfrac]5/15[/ggfrac]. Next, you see that [ggfrac]1/3[/ggfrac] has a denominator of 3, so you multiply the numerator and denominator of the other fraction by 3: [ggfrac]2/5[/ggfrac] × [ggfrac]3/3[/ggfrac]= [ggfrac]6/15[/ggfrac]. Rewrite the original addition as [ggfrac]5/15[/ggfrac]+ [ggfrac]6/15[/ggfrac]= [ggfrac]11/15[/ggfrac]. Try this one yourself. Add [ggfrac]1/6[/ggfrac] + [ggfrac]3/4[/ggfrac] by finding a common denominator using the faster way.

### Guinea Pig Weights

Let's say that you have two tiny guinea pigs. One weighs [ggfrac]4/5[/ggfrac] of a pound and the other weighs [ggfrac]2/3[/ggfrac] of a pound. You want to figure out how much more one guinea pig weighs than the other. You can find this difference using the equation [ggfrac]4/5[/ggfrac] - [ggfrac]2/3[/ggfrac]= ?. First, find equivalent fractions that have a common denominator. The second fraction has a denominator of 3, so multiply the numerator and denominator of the first fraction by 3. [ggfrac]4/5[/ggfrac] × [ggfrac]3/3[/ggfrac] = [ggfrac]12/55[/ggfrac]. Next, multiply [ggfrac]2/3[/ggfrac] × [ggfrac]5/5[/ggfrac] since 5 is the denominator of [ggfrac]4/5[/ggfrac]. [ggfrac]2/3[/ggfrac] × [ggfrac]5/5[/ggfrac]= [ggfrac]10/15[/ggfrac]. Rewrite the original equation [ggfrac]4/5[/ggfrac] - [ggfrac]2/3[/ggfrac] as [ggfrac]12/15[/ggfrac] - [ggfrac]10/15[/ggfrac]= [ggfrac]2/15[/ggfrac]. One guinea pig weighs [ggfrac]2/15[/ggfrac] of a pound more than the other guinea pig. Try this one yourself. My bag of candy weighs [ggfrac]3/4[/ggfrac] of a pound and my friend's bag of candy weighs [ggfrac]2/3[/ggfrac] of a pound. How much more does my bag of candy weigh?

Guinea Pig Weights Let's say that you have two tiny guinea pigs. One weighs [ggfrac]4/5[/ggfrac] of a pound and the other weighs [ggfrac]2/3[/ggfrac] of a pound. You want to figure out how much more one guinea pig weighs than the other. You can find this difference using the equation [ggfrac]4/5[/ggfrac] - [ggfrac]2/3[/ggfrac]= ?. First, find equivalent fractions that have a common denominator. The second fraction has a denominator of 3, so multiply the numerator and denominator of the first fraction by 3. [ggfrac]4/5[/ggfrac] × [ggfrac]3/3[/ggfrac] = [ggfrac]12/55[/ggfrac]. Next, multiply [ggfrac]2/3[/ggfrac] × [ggfrac]5/5[/ggfrac] since 5 is the denominator of [ggfrac]4/5[/ggfrac]. [ggfrac]2/3[/ggfrac] × [ggfrac]5/5[/ggfrac]= [ggfrac]10/15[/ggfrac]. Rewrite the original equation [ggfrac]4/5[/ggfrac] - [ggfrac]2/3[/ggfrac] as [ggfrac]12/15[/ggfrac] - [ggfrac]10/15[/ggfrac]= [ggfrac]2/15[/ggfrac]. One guinea pig weighs [ggfrac]2/15[/ggfrac] of a pound more than the other guinea pig. Try this one yourself. My bag of candy weighs [ggfrac]3/4[/ggfrac] of a pound and my friend's bag of candy weighs [ggfrac]2/3[/ggfrac] of a pound. How much more does my bag of candy weigh?