How to Increase and Reduce the Terms of Fractions

When you cut a cake into two pieces and take one piece, you have 1/2 of the cake. And when you cut it into four pieces and take two, you have 2/4 of the cake. Finally, when you cut it into six pieces and take three, you have 3/6 of the cake. Notice that in all these cases, you get the same amount of cake.

This shows you that the fractions 1/2, 2/4, and 3/6 are equal; so are the fractions 10/20 and

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Most of the time, writing this fraction as 1/2 is preferred because the numerator and denominator are the smallest possible numbers. In other words, the fraction 1/2 is written in lowest terms. At the end of a problem, you often need to reduce a fraction, or write it in lowest terms. There are two ways to do this — the informal way and the formal way:

  • The informal way to reduce a fraction is to divide both the numerator and the denominator by the same number.

    Advantage: The informal way is easy.

    Disadvantage: It doesn’t always reduce the fraction to lowest terms (though you do get the fraction in lowest terms if you divide by the greatest common factor).

  • The formal way is to decompose both the numerator and the denominator into their prime factors and then cancel common factors.

    Advantage: The formal way always reduces the fraction to lowest terms.

    Disadvantage: It takes longer than the informal way.

Start every problem using the informal way. If the going gets rough and you’re still not sure whether your answer is reduced to lowest terms, switch over to the formal way.

Sometimes at the beginning of a fraction problem, you need to increase the terms of a fraction — that is, write that fraction using a greater numerator and denominator. To increase terms, multiply both the numerator and denominator by the same number.

Sample questions

  1. Increase the terms of the fraction 4/5 to a new fraction whose denominator is 15:

    image1.jpg

    To start out, write the problem as follows:

    image2.jpg

    The question mark stands for the numerator of the new fraction, which you want to fill in. Now divide the larger denominator (15) by the smaller denominator (5).

    15 / 5 = 3

    Multiply this result by the numerator:

    3 x 4 = 12

    Finally, take this number and use it to replace the question mark:

    image3.jpg
  2. Reduce the fraction 18/42 to lowest terms.

    image4.jpg

    The numerator and denominator aren’t too large, so use the informal way: To start out, try to find a small number that the numerator and denominator are both divisible by. In this case, notice that the numerator and denominator are both divisible by 2, so divide both by 2:

    image5.jpg

    Next, notice that the numerator and denominator are both divisible by 3, so divide both by 3:

    image6.jpg

    At this point, there’s no number (except for 1) that evenly divides both the numerator and denominator, so this is your answer.

  3. Reduce the fraction 135/196 to lowest terms.

    image7.jpg

    The numerator and denominator are both over 100, so use the formal way. First, decompose both the numerator and denominator down to their prime factors:

    image8.jpg

    The numerator and denominator have no common factors, so the fraction is already in lowest terms.

Practice questions

  1. Increase the terms of the fraction 2/3 so that the denominator is 18.

  2. Increase the terms of 4/9, changing the denominator to 54.

  3. Reduce the fraction 12/60 to lowest terms.

  4. Reduce 45/75 to lowest terms.

  5. Reduce the fraction 135/180 to lowest terms.

  6. Reduce 108/217 to lowest terms.

Following are the answers to the practice questions:

  1. image9.jpg

    To start out, write the problem as follows:

    image10.jpg

    Divide the larger denominator (18) by the smaller denominator (3) and then multiply this result by the numerator (2):

    6 x 2 = 12

    Take this number and use it to replace the question mark; your answer is

    image11.jpg
  2. image12.jpg

    Write the problem as follows:

    image13.jpg

    Divide the larger denominator (54) by the smaller denominator (9) and then multiply this result by the numerator (4):

    6 x 4 = 24

    Take this number and use it to replace the question mark; your answer is

    image14.jpg
  3. image15.jpg

    The numerator (12) and the denominator (60) are both even, so divide both by 2:

    image16.jpg

    They’re still both even, so divide both by 2 again:

    image17.jpg

    Now the numerator and denominator are both divisible by 3, so divide both by 3:

    image18.jpg
  4. image19.jpg

    The numerator (45) and the denominator (75) are both divisible by 5, so divide both by 5:

    image20.jpg

    Now the numerator and denominator are both divisible by 3, so divide both by 3:

    image21.jpg
  5. image22.jpg

    The numerator (135) and the denominator (180) are both divisible by 5, so divide both by 5:

    image23.jpg

    Now the numerator and denominator are both divisible by 3, so divide both by 3:

    image24.jpg

    They’re still both divisible by 3, so divide both by 3 again:

    image25.jpg
  6. image26.jpg

    With a numerator and denominator this large, reduce using the formal way. First, decompose both the numerator and denominator down to their prime factors:

    image27.jpg

    The numerator and denominator have no common factors, so the fraction is already in lowest terms.

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