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Fractions – Utilizing the Four Math Operations

Learning fractions can be quite complicated. All fractions have a top number (numerator) and a bottom number (denominator). There are problems involving fractions that require several steps before arriving at the solution. Many fraction problems also require the use of more than one basic mathematical operation. The four operations are addition, subtraction, multiplication and division. If you lack proficiency in any of these areas, you will have trouble doing fractions. Mastering fractions takes a lot of practice. In this article, I will present various examples to demonstrate how the four mathematical operations come into play with solving fractions.

Example 1: Adding Fractions (Same Denominator)

5/9 + 2/9 = 7/9

When you add five ninths and two ninths, you simply add the numerators of 5 and 2, which become 7. The denominator of 9 remains the same. The answer is seven ninths.

Example 2: Adding fractions (same denominator and reduced to simplest form)

5/10 + 3/10 = 8/10 = 4/5

The numerators add up to 8. The denominator remains at 10. The answer is eight tenths. However, eight tenths can be reduced to a smaller equivalent fraction. You need to determine the highest number (common factor) that can be divided equally between the numerator and the denominator. Here, 8 and 10 can be divided by 2. Eight tenths can be converted to four fifths, which is the final answer.

Example 3: Adding Fractions (Different Denominator and Reduced to Simplest Form)

4/8 + 3/12 = 12/24 + 6/24 = 18/24 = 3/4

The two denominators must be converted into the same denominator before you can add. The denominators here are 8 and 12. First, you need to figure out the lowest number in which 8 and 12 can be multiplied equally. The lowest number would be 24. You then need to convert 4/8 and 3/12 into fractions that will have 24 as the denominator. For 4/8, you would multiply both numbers by 3 to get 12/24. For 3/12, you would multiply both numbers by 2 to get 6/24. You will then add 12/24 and 6/24 to get 18/24. The 18/24 must now be reduced to the simplest form. The highest common factor for 18 and 24 is 6. 18/24 divided by 6 equals 3/4.

Example 4: Subtracting Fractions (Same Denominator and Reduced to Simplest Form)

18/25 – 8/25 = 10/25 = 2/5

The numerators are subtracted from 10. The denominator remains 25. 10/25 can be further reduced. 10 and 25 can both be divided by 5. The final answer is two fifths.

Example 5: Subtracting a fraction (different denominators and reduced to the simplest form)

30/40 – 15/60 = 90/120 – 30/120 = 60/120 = 1/2

Both denominators must be converted to the same denominator before you can subtract. The denominators here are 40 and 60. Determine the lowest number in which 40 and 60 can be multiplied equally. The lowest number is 120. You then need to convert both 30/40 and 15/60 into fractions that will have 120 as the denominator. For 30/40, you would multiply both numbers by 3 to get 90/120. For 15/60, you would multiply both numbers by 2 to get 30/120. You can now subtract 90/120 and 30/120 to get 60/120. 60 and 120 can each be divided by 60. 60/120 becomes 1/2 which is the final answer.

Example 6: Multiplying Fractions (Simple Problem)

7/8 x 3/4 = 21/32

Simply multiply the numerators and denominators to get the answer.

Example 7: Multiplying Fractions (reduced to simplest form – cross-cancellation)

15/ 25 x 5/30 = 1/5 x 1/2 = 1/10

Both fractions can be reduced to the simplest form by canceling each other’s numerator and denominator. The numerator of the first fraction (15) and the denominator of the second fraction (30) are both divisible by 15. “15” becomes 1 and “30” becomes 2. The same is true for the numerator of the second fraction (5) and the denominator of the first fraction (25). “5” becomes 1 and “25” becomes 5. You now multiply 1/5 and 1/2. Multiply the numerators and denominators. The final answer is 1/10.

Example 8: Division of fractions (simple problem)

5/9 / 7/11 = 5/9 x 11/7 = 55/63

When fractions are divided, you must “reverse” the second fraction and change the sign of operation from division to multiplication. 7/11 now becomes 11/7. You are now going to multiply the fractions.

Example 9: Division of Fractions (reduced to simplest form)

3/9 / 7/8 = 3/9 x 8/7 = 24/63 = 8/21

Flip 7/8 to 8/7 and change the division sign to multiplication. Multiply the fractions. 24/63 can be further reduced. 24 and 63 are both divisible by 3 (greatest common factor). The final answer is 8/21.

Example 10: Division of fractions (reduced to simplest form – cross-cancellation)

36/45 / 18/15 = 36/45 x 15/18 = 2/3 x 1/1 = 2/3

Flip 18/15 to 15/18 and change the division sign to multiplication. 36/45 and 15/18 can be reduced by cross-cancellation. The numerator of the first fraction (36) and the denominator of the second fraction (18) are both divisible by 18. “36” becomes 2 and “18” becomes 1. Cross-cancellation is now performed for the numerator of the second fraction (15) and the denominator of the first fraction (45). “15” becomes 1 and “45” becomes 3. You now multiply 2/3 and 1/1. The final answer is 2/3.

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