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Making Equivalent Fractions with One

We'll be making a lot of "like fractions" in this section (fractions with common denominators). Remember that 1 can be represented by a fraction when the numerator and denominator are the same value. 2/2 is the same as 1. 9/9 is the same as 1. 52/52 is the same as one. If that is confusing, think of it as a division problem. 2÷2=1. 9÷9=1. 52÷52=1. Also, remember that in multiplication anything multiplied by 1 is the same value. 2*1=2. 9*1=9. 52*1=52. That math fact is called the identity property of multiplication. We're going to use this trick to make like fractions.

We know that 1/3 * 1 = 1/3. Let's say our fraction problem needed the solution to have the denominator 18 (bottom number). Use the concept that 1 is equivalent to 6/6. That means...

• Start: 1/3 * 1 = 1/3
• Swap: 1/3 * 6/6 = 1/3
• Multiply the Fractions: (1*6)/(3*6) = 6/18
• Simplify to Check Answer: 6/18 = 1/3

We used the identity property to create equivalent fractions. We created the same denominator for all of our terms.

Comparing Fractions

You will get a lot of problems where you are asked to compare fractions. Is 1/2 bigger or smaller than 1/3? You should already know about "greater than" and "less than" symbols.

It's easier with whole numbers...
• Compare 2 and 1. You know that two is greater than one.
• Compare 13 and 27. You know that thirteen is less than twenty-seven.
• Compare -40 and -2. We have worked with negative integers before. -40 is less than -2.

So what about fractions? One some levels it's just as easy. Fractions with larger denominators (bottom number) have more pieces that are possible. When you have more pieces that are possible in the same space, the pieces have to be smaller. If the number of pieces (numerator) in each fraction is the same, the one with the larger denominator will always be less than the other. This only works when you can compare the same number of pieces.

Examples:
Compare 1/2 and 1/5.
Think about a pie. One pie is cut into two pieces and one is cut into five pieces. Which piece is bigger? Half of a pie is bigger than one fifth of a pie. So 1/2 is greater than 1/5.

Compare 5/8 and 5/10.
Start by noticing that you have five pieces of each. Since they are the same number, we can ignore them. Then look at the denominators and think about pieces of a pie. An eighth of a pie is bigger than a tenth of a pie. Basically, you have five bigger pieces compared to five smaller pieces. So 5/8 is greater than 5/10.

When the numerators are the same, we don't have to worry about converting any numbers. Let's look at like fractions (same denominators). They are easy. You only need to focus on the values of the numerators without converting anything.

Examples:
Compare 2/9 and 6/9.
You have the same denominators, so the size of the pieces is the same. Now look up to the numerators. Two pieces compared to six pieces. You have this one. If 2 < 6 then...
2/9 < 6/9

Compare 8/17 to 3/17
Once again, you have the same denominators. The pieces are the same size. Compare eight to three. Since eight is greater than three...
8/17 > 3/17

The easy ones are out of the way now. But what happens when you have unlike fractions (different denominators) with different numerators? You are going to need to make them "like fractions" to really compare them. That means you will need the same bottom numbers (common denominators) for each fraction. You're going to need a little multiplication to do this one.

Examples:
Compare 5/6 and 17/18
We have sixths and eighteenths for denominators. We need to make them like fractions. They have the common factor of 6 (6x3=18). That's good, we only have to deal with the 5/6 term. The 17/18 can stay the way it is. Since we know that 6x3=18, let's multiply the numerator and the denominator by 3. Use the start-swap-multiply process from above.
5/6 = 5/6 * 1 = 5/6 * 3/3 = (5*3)/(6*3) = 15/18
Now you can compare 15/18 and 17/18. No problem.
15/18 < 17/18

Compare 6/9 and 3/4.
Notice that we have ninths and fourths for denominators. There are no common factors on this problem. The fast way is to create equivalent fractions for each term and compare them. How? Multiply the first term by 4/4 and the second by 9/9. In other words, we will be multiplying both the top and bottom numbers of one term by the denominator of the other. Use the start-swap-multiply process from above for both terms.

6/9 = 6/9 * 1 = 6/9 * 4/4 = (6*4)/(9*4) = 24/36
3/4 = 3/4 * 1 = 3/4 * 9/9 = (3*9)/(4*9) = 27/36

Did you see that? When you multiply by the denominator of the other term, you wind up with like fractions. Now we can compare 24/36 and 27/36. Easy as pie.
24/36 < 27/36

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