Equivalence of '1/b of c' and 'c × 1/b'
Equivalence of 1/b of c and c × 1/b is a Grade 5 math skill from Illustrative Mathematics Chapter 2 (Fractions as Quotients and Fraction Multiplication) that establishes the three-way equivalence: (1/b) of c = c ÷ b = c × (1/b). The word "of" indicates multiplication, and multiplying by a unit fraction equals dividing by its denominator. This foundational equivalence connects fraction language, division, and multiplication.
Key Concepts
Finding a unit fraction of a whole number is equivalent to multiplying the whole number by the unit fraction. The word "of" in this context implies multiplication.
$$\frac{1}{b} \text{ of } c = c \div b = c \times \frac{1}{b}$$.
Common Questions
What does it mean to find 1/b of c?
Finding 1/b of c means dividing c into b equal groups and taking one group. This equals c ÷ b, which also equals c × (1/b). For example, 1/4 of 12 = 12 ÷ 4 = 12 × (1/4) = 3.
Why does finding a unit fraction of a number equal multiplying by that fraction?
The word of signals multiplication. Finding 1/b of c means taking one of b equal parts of c. This is the same as c ÷ b (dividing into b parts) and c × (1/b) (multiplying by the unit fraction) — all three are equivalent.
What chapter covers the equivalence of fraction of and multiplication in Illustrative Mathematics Grade 5?
The equivalence of 1/b of c and c × 1/b is covered in Chapter 2 of Illustrative Mathematics Grade 5, titled Fractions as Quotients and Fraction Multiplication.
What are examples of the 1/b of c equivalence?
(1/2) of 8 = 8 ÷ 2 = 8 × (1/2) = 4. (1/4) of 12 = 12 ÷ 4 = 12 × (1/4) = 3. The three expressions always give the same result.
How does this equivalence help solve fraction word problems?
When a problem says find 1/3 of 15, you know you can either divide 15 by 3 or multiply 15 by 1/3. Choose whichever is easier mentally: 15 ÷ 3 = 5, or 15 × (1/3) = 5.