Equivalent Fractions: Fifth Grade Mathematics

Equivalent fractions represent the identical value or amount of a whole, even though they are expressed using different numerators and denominators.

The Principle of Equivalent Fractions

Fractions are equivalent when they occupy the exact same point on a number line and cover the exact same area of a unit whole.

+--------------------------------------------------------------------------+
|                  EQUIVALENT FRACTION AREA REPRESENTATIONS                |
|                                                                          |
|   1/2 :  [============|------------]   (1 of 2 equal parts shaded)       |
|                                                                          |
|   2/4 :  [======|======|------|------] (2 of 4 equal parts shaded)       |
|                                                                          |
|   4/8 :  [===|===|===|===|---|---|---|---] (4 of 8 equal parts shaded)   |
|                                                                          |
|   All three models have the exact same total shaded area!                |
+--------------------------------------------------------------------------+

The Identity Property Rule (Multiplying by 1)

To generate an equivalent fraction, multiply (or divide) both the numerator and the denominator by the same non-zero whole number. Because n/n equals 1, multiplying by n/n does not change the fraction's actual value.

Generating Equivalent Fractions by Multiplying by 1:
    3       3 * 4       12
   ---  *  -------  =  ----
    5       5 * 4       20

   (Multiplying by 4/4 is multiplying by 1 whole!)

Creating Equivalent Fractions by Dividing

You can also create equivalent fractions with smaller numbers by dividing both terms by a common factor:
- (18 / 6) / (24 / 6) = 3/4.

- This reverse process is called simplifying or reducing.

Exercises and Solutions

Exercise 1: Generating Equivalent Fractions. Generate three different fractions equivalent to 4/7 by multiplying the numerator and denominator by 2, 3, and 5.

Solution 1: Multiply 4/7 by 2/2: (4 * 2) / (7 * 2) = 8/14. Multiply 4/7 by 3/3: (4 * 3) / (7 * 3) = 12/21. Multiply 4/7 by 5/5: (4 * 5) / (7 * 5) = 20/35. The three equivalent fractions are 8/14, 12/21, and 20/35.

Exercise 2: Finding a Missing Term. Find the unknown numerator n in the equivalent fraction equation: 5/8 = n/56. Explain the mathematical step used to solve for n.

Solution 2: Examine how the denominator changed: the original denominator 8 was multiplied by 7 to become 56 (56 / 8 = 7). To keep the fractions equivalent, the numerator must also be multiplied by the identical factor of 7. Multiply the numerator: 5 * 7 = 35. Therefore, n = 35, and the equivalent fraction is 35/56.