Chapter 2, Problem 1P

Work with fractions.

1. (a) Write as a mixed number $\frac{31}{4}$.
2. (b) Write as an improper fraction $3\frac{7}{8}$.

   Write as an equivalent fraction.

3. (c) $\frac{5}{16}=\frac{?}{64}$
4. (d) $1\frac{3}{4}=\frac{?}{32}$
5. (e) Write in lowest terms $\frac{10}{64}$.
6. (f) Which is larger, $1\frac{7}{8}$ or $\frac{5}{3}$?

To determine

To write: The improper fraction $\frac{31}{4}$ as a mixed number.

Answer to Problem 1P

The mixed number of the fraction $\frac{31}{4}$ is $\underset{\_}{7\frac{3}{4}}$.

Explanation of Solution

Procedure used:

Mixed number is the sum of a whole number and a proper fraction.

To convert an improper fraction to a mixed number, divide the numerator by its denominator and obtain the quotient and the remainder.

Then, the mixed numeral can be written as $\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}$, where the quotient is a whole number, remainder and divisor represents a fraction.

Calculation:

The given fraction is $\frac{31}{4}$.

Here, the numerator is 31 and the denominator is 4.

To write the improper fraction as a mixed numeral, divide the numerator by the denominator as follows.

$4\begin{array}{c}\hfill 7\\ \hfill \overline{)\begin{array}{l}31\\ \underset{\_}{28}\\ \text{ 3}\end{array}}\end{array}$

Note that the quotient is 4 and the remainder is 3.

By the above mentioned procedure, the mixed number is, $\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}=7\frac{3}{4}$.

Thus, the mixed number of the improper fraction $\frac{31}{4}$ is $\underset{\_}{7\frac{3}{4}}$.

To determine

To write: The mixed number $3\frac{7}{8}$ as an improper fraction.

Answer to Problem 1P

The improper fraction of $3\frac{7}{8}$ is $\underset{\_}{\frac{\text{31}}{8}}$.

Explanation of Solution

Procedure used:

1) Identify the quotient, remainder and divisor from the mixed number which is in the form of $\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}$.

Multiply the quotient and the divisor (original denominator).

2) Add the number obtained from step 1 with the remainder which gives a new numerator.

3) The improper fraction is, $\frac{\text{The new numerator}}{\text{The original denominator}}$.

Calculation:

The given mixed number is $3\frac{7}{8}$.

Step 1:

Here, 3 is the quotient, 7 is the remainder and 8 is the divisor.

Multiply the quotient 3 and the divisor 8.

That is, $3\times 8=24$.

Step 2:

Add the number 24 with the remainder 7.

That is, $24+7=31$.

Thus, 7 is the new numerator.

Step 3:

Thus, the improper fraction is $\frac{\text{The new numerator}}{\text{The original denominator}}=\frac{\text{31}}{8}$.

Thus, the improper fraction of $3\frac{7}{8}$ is $\underset{\_}{\frac{\text{31}}{8}}$.

To determine

To write: An equivalent fraction $\frac{5}{16}=\frac{?}{64}$.

Answer to Problem 1P

The completed equivalent fraction is $\underset{\_}{\frac{5}{16}=\frac{20}{64}}$.

Explanation of Solution

Definition used:

Equivalent fraction:

Two or more fraction having the same values are said to be equivalent fractions.

Calculation:

The given complete fraction is $\frac{5}{16}$ and the incomplete fraction is $\frac{?}{64}$.

Here, the numerator is 5 and the denominator is 16 for the complete fraction.

Rewrite the fraction $\frac{5}{16}$ as an equivalent fraction with denominator equal to 64 as follows.

$\frac{5\times \boxed{}}{16\times \boxed{}}=\frac{?}{64}$ (1)

List the factors of 64 as tabulated below in Table 1.

Factors of 64:

1, 64

2, 32

4, 16

8, 8

Since, the denominator of the given fraction is 16, the third row of the above can satisfies the incomplete fraction.

Fill the number 4 in the empty boxes of equation (1) as follows.

$\frac{5\times \boxed{\text{4}}}{16\times \boxed{\text{4}}}=\frac{20}{64}$

Thus, the completed equivalent fraction is $\underset{\_}{\frac{5}{16}=\frac{20}{64}}$.

To determine

To complete: The equivalent fraction $1\frac{3}{4}=\frac{?}{32}$.

Answer to Problem 1P

The completed equivalent fraction is $\underset{\_}{1\frac{3}{4}=\frac{56}{32}}$.

Explanation of Solution

Procedure used:

1) Identify the quotient, remainder and divisor from the mixed number which is in the form of $\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}$.

Multiply the quotient and the divisor (original denominator).

2) Add the number obtained from step 1 with the remainder which gives a new numerator.

3) The improper fraction is, $\frac{\text{The new numerator}}{\text{The original denominator}}$.

Calculation:

The given complete mixed fraction is $1\frac{3}{4}$ and the incomplete fraction is $\frac{?}{16}$.

First convert the mixed fraction $1\frac{3}{4}$ to improper fraction.

Step 1:

Here, 1 is the quotient, 3 is the remainder and 4 is the divisor.

Multiply the quotient 1 and the divisor 4.

That is, $1\times 4=4$.

Step 2:

Add the number 4 with the remainder 3.

That is, $4+3=7$.

Thus, 7 is the new numerator.

Step 3:

Thus, the improper fraction is $\frac{\text{The new numerator}}{\text{The original denominator}}=\frac{\text{7}}{4}$.

Thus, the improper fraction of $1\frac{3}{4}$ is $\frac{\text{7}}{4}$.

Here, the numerator is 7 and the denominator is 4 for the complete fraction.

Rewrite the fraction $\frac{\text{7}}{4}$ as an equivalent fraction with denominator equal to 32 as follows.

$\frac{7\times \boxed{}}{4\times \boxed{}}=\frac{?}{32}$ (2)

List the factors of 16 as tabulated below in Table 2.

Factors of 32:

1, 32

2, 16

4, 8

Since, the denominator of the given fraction is 4, the third row of the above can satisfies the incomplete fraction.

Fill the number 8 in the empty boxes of equation (2) as follows.

$\frac{7\times \boxed{\text{8}}}{4\times \boxed{\text{8}}}=\frac{56}{32}$

Thus, the completed equivalent fraction is $\underset{\_}{1\frac{3}{4}=\frac{56}{32}}$.

To determine

To write: The lowest term of the fraction $\frac{10}{64}$.

Answer to Problem 1P

The lowest term of the fraction $\frac{10}{64}$ is $\underset{\_}{\frac{5}{32}}$.

Explanation of Solution

The given fraction is $\frac{10}{64}$.

Here, the numerator is 10 and the denominator is 64.

Step 1:

The prime factorization of 10 is $2\cdot 5$ and 64 is $2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2$.

The common prime is 2.

Step 2:

Since the common prime is 2, no need to do the multiplication.

Step 3:

Divide the numerator and the denominator of the fraction $\frac{10}{64}$ by 2 as follows.

$\frac{10÷2}{64÷2}=\frac{5}{32}$

Thus, the lowest term of the fraction $\frac{42}{64}$ is $\underset{\_}{\frac{5}{32}}$.

To determine

The largest fraction among the fractions $1\frac{7}{8}\text{ and }\frac{5}{3}$.

Answer to Problem 1P

The fraction $\underset{\_}{1\frac{7}{8}}$ is larger than the fraction $\frac{5}{3}$.

Explanation of Solution

The given fractions are $1\frac{7}{8}\text{ and }\frac{5}{3}$.

Convert the mixed fraction $1\frac{7}{8}$ to improper fraction as follows.

Here, 1 is the quotient, 7 is the remainder and 8 is the divisor.

Multiply the quotient 1 and the divisor 8.

That is, $1\times 8=8$.

Add the number 8 with the remainder 7.

That is, $8+7=15$.

Thus, 15 is the new numerator.

The improper fraction is $\frac{\text{The new numerator}}{\text{The original denominator}}=\frac{\text{15}}{8}$.

Thus, the improper fraction of $1\frac{7}{8}$ is $\frac{\text{15}}{8}$.

Now the fractions are $\frac{\text{15}}{8}\text{ and }\frac{5}{3}$.

Compute the equivalent fraction for the given fraction with same denominator.

The denominators of the given fractions are 8 and 3.

The prime factorization of 8 is $2\cdot 2\cdot 2$ and 3 is 3.

The G.C.D of 8 and 3 is 24.

Multiply and divide the fraction $\frac{\text{15}}{8}$ by 3 as follows.

$\begin{array}{c}\frac{\text{15}}{8}=\frac{15\times 3}{8\times 3}\\ =\frac{45}{24}\end{array}$

Similarly multiply and divide the fraction $\frac{5}{3}$ by 8 as follows.

$\begin{array}{c}\frac{5}{3}=\frac{5\times 8}{3\times 8}\\ =\frac{40}{24}\end{array}$

Compare the new fractions and obtain the larger fraction among them as follows.

$\begin{array}{l}\frac{45}{24}>\frac{40}{24}\\ \frac{\text{15}}{8}>\frac{5}{3}\\ 1\frac{7}{8}>\frac{5}{3}\end{array}$

Thus, the fraction $\underset{\_}{1\frac{7}{8}}$ is larger than the fraction $\frac{5}{3}$.