Finding a Fraction of a Whole Number: Fifth Grade Mathematics

Finding a fraction of a whole number interprets the word "of" as multiplication, meaning you divide into equal groups and take a specified number of those groups.

The Mathematical Meaning of "Of"

In mathematics, the word "of" between a fraction and a whole number indicates multiplication. For example, "1/2 of 12" means 1/2 * 12 = 6.

+--------------------------------------------------------------------------+
|                  FINDING A FRACTION OF A NUMBER: 3/4 OF 20               |
|                                                                          |
|       Step 1: Divide the whole number into equal groups matching the     |
|               denominator (4 groups):                                    |
|               20 / 4 = 5 items per group                                 |
|                                                                          |
|       [ Group 1: 5 ]  [ Group 2: 5 ]  [ Group 3: 5 ]  [ Group 4: 5 ]     |
|       <------------------- Take 3 Groups ------------------>             |
|                                                                          |
|       Step 2: Multiply by the numerator (3 groups):                      |
|               3 * 5 = 15                                                 |
|                                                                          |
|       Therefore, 3/4 of 20 = 15.                                         |
+--------------------------------------------------------------------------+

Two Computational Approaches

You can find a fraction of a number in either order:
- Divide first, then multiply: (Whole / Denominator) * Numerator. For 3/5 of 40: (40 / 5) * 3 = 8 * 3 = 24.

- Multiply first, then divide: (Whole * Numerator) / Denominator. (40 * 3) / 5 = 120 / 5 = 24.

- Both methods yield the identical answer. Dividing first keeps numbers smaller and easier to compute mentally.

Bar Modeling Fractions of Sets

Tape diagrams provide clear visual proofs for fraction-of-a-number scenarios.

Bar Model: 2/3 of $45

  Total: [================== $45 ==================]
  Units: [   $15   ][   $15   ][   $15   ]
         <--- 2 units = $30 --->

Exercises and Solutions

Exercise 1: Mental Computation. Calculate 4/7 of 35 using the divide-first method. Show both intermediate steps.

Solution 1: Step 1: Divide the whole number 35 by the denominator 7: 35 / 7 = 5. This tells you that each seventh is worth 5. Step 2: Multiply that unit amount by the numerator 4: 5 * 4 = 20. Therefore, 4/7 of 35 is 20.

Exercise 2: Class Survey Word Problem. A fifth-grade class has 28 students. If 3/4 of the students participate in an after-school music or sports program, how many students participate, and how many do not?

Solution 2: Find 3/4 of 28: divide 28 by 4 = 7; multiply 7 by 3 = 21 students participate. To find how many students do not participate, subtract 21 from 28: 28 - 21 = 7 students (or calculate 1/4 of 28 = 7). Thus, 21 students participate and 7 students do not participate.