Question

1. find the value of each expression. show your reasoning and check your answer.

a. $2\frac{1}{7} \div \frac{2}{7}$
b. $\frac{17}{20} \div \frac{1}{4}$

Explanation:

Part a: \( 2\frac{1}{7} \div \frac{2}{7} \)

Step 1: Convert mixed number to improper fraction

A mixed number \( 2\frac{1}{7} \) can be converted to an improper fraction by multiplying the whole number (2) by the denominator (7) and adding the numerator (1). So, \( 2\frac{1}{7}=\frac{2\times7 + 1}{7}=\frac{14 + 1}{7}=\frac{15}{7} \).

Step 2: Divide by multiplying by reciprocal

Dividing by a fraction \( \frac{2}{7} \) is the same as multiplying by its reciprocal \( \frac{7}{2} \). So, \( \frac{15}{7}\div\frac{2}{7}=\frac{15}{7}\times\frac{7}{2} \).

Step 3: Simplify the expression

The 7 in the numerator and denominator cancels out, leaving \( \frac{15}{2} \), which is equal to \( 7\frac{1}{2} \) or 7.5.

Step 4: Check the answer

To check, we can multiply the quotient \( \frac{15}{2} \) by the divisor \( \frac{2}{7} \). \( \frac{15}{2}\times\frac{2}{7}=\frac{15\times2}{2\times7}=\frac{15}{7}=2\frac{1}{7} \), which matches the original dividend. So the answer is correct.

Step 1: Divide by multiplying by reciprocal

Dividing by \( \frac{1}{4} \) is the same as multiplying by its reciprocal \( 4 \) (or \( \frac{4}{1} \)). So, \( \frac{17}{20}\div\frac{1}{4}=\frac{17}{20}\times4 \).

Step 2: Simplify the expression

We can simplify 4 and 20 by dividing both by 4. 4 ÷ 4 = 1 and 20 ÷ 4 = 5. So the expression becomes \( \frac{17}{5}\times1=\frac{17}{5} \), which is equal to \( 3\frac{2}{5} \) or 3.4.

Step 3: Check the answer

To check, we multiply the quotient \( \frac{17}{5} \) by the divisor \( \frac{1}{4} \). \( \frac{17}{5}\times\frac{1}{4}=\frac{17}{20} \), which matches the original dividend. So the answer is correct.

Answer:

\( \frac{15}{2} \) (or \( 7\frac{1}{2} \) or 7.5)

Part b: \( \frac{17}{20}\div\frac{1}{4} \)