Question

1. how many groups of \\(\frac{5}{6}\\) are in each of the following quantities? show your thinking.\

a \\(\frac{2}{3}\\)\
b \\(1\frac{1}{2}\\)\
c \\(4\frac{1}{6}\\)

Explanation:

Part a:

Step 1: Recall the division rule

To find how many groups of a number \( b \) are in a number \( a \), we use division: \( a\div b \). Here, \( a = \frac{2}{3} \) and \( b=\frac{5}{6} \), so we calculate \( \frac{2}{3}\div\frac{5}{6} \).

Step 2: Divide the fractions

Dividing by a fraction is the same as multiplying by its reciprocal: \( \frac{2}{3}\times\frac{6}{5} \). Simplify the numerators and denominators: \( \frac{2\times6}{3\times5}=\frac{12}{15}=\frac{4}{5} \).

Part b:

Step 1: Convert mixed number to improper fraction

The mixed number \( 1\frac{1}{2} \) is equal to \( \frac{1\times2 + 1}{2}=\frac{3}{2} \). Now we need to find \( \frac{3}{2}\div\frac{5}{6} \).

Step 2: Divide the fractions

Multiply by the reciprocal: \( \frac{3}{2}\times\frac{6}{5} \). Simplify: \( \frac{3\times6}{2\times5}=\frac{18}{10}=\frac{9}{5}=1\frac{4}{5} \).

Part c:

Step 1: Convert mixed number to improper fraction

The mixed number \( 4\frac{1}{6} \) is equal to \( \frac{4\times6+ 1}{6}=\frac{25}{6} \). Now we find \( \frac{25}{6}\div\frac{5}{6} \).

Step 2: Divide the fractions

Multiply by the reciprocal: \( \frac{25}{6}\times\frac{6}{5} \). The 6s cancel out, and we have \( \frac{25}{5}=5 \).

Answer:

a. \(\boldsymbol{\frac{4}{5}}\)
b. \(\boldsymbol{1\frac{4}{5}}\)
c. \(\boldsymbol{5}\)