Chapter 1 Use Positive Rational Numbers
Section 1.5: Divide Fractions by Fractions
Page 37 Exercise 1 Answer
To answer the question write the fraction \(\frac{2}{3}\) as a fraction with the denominator 6.
\(\frac{2}{3}=\frac{2 \times 2}{3 \times 2}\)
= \(\frac{4}{6}\)
Since \(\frac{2}{3}\) of the original bar remains, and \(\frac{2}{3}\) is equal to \(\frac{4}{6}\), which means there are four \(\frac{1}{6}\) parts left.
\(\frac{2}{3} \div \frac{1}{6}=\frac{2}{3} \times \frac{6}{1}\)
= \(\frac{12}{3}\)
= 4
Result
4 parts
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 37 Exercise 1 Answer
In the Solve & Discuss It! problem, we found that \(\frac{2}{3} \div \frac{1}{6}=4\).
To check our answer using multiplication, we can use a related multiplication fact. Recall that a division statement a ÷ b = c has a related multiplication fact of a = b × c. For the division statement \(\frac{2}{3}\) ÷ \(\frac{1}{6}\) = 4, the related multiplication fact is \(\frac{2}{3}\) = \(\frac{1}{6}\) × 4.
Our answer is then correct if \(\frac{1}{6}\) × 4 is equal to \(\frac{2}{3}\). Multiplying \(\frac{1}{6}\) and 4 gives:
\(\frac{1}{6} \times 4=\frac{1}{6} \times \frac{4}{1}=\frac{1 \times 4}{6 \times 1}=\frac{4}{6}=\frac{2}{3}\)
Our answer is then correct.
Result
We can use the related multiplication fact \(\frac{2}{3}\) = \(\frac{1}{6}\) x 4 to check our answer.
Page 38 Exercise 1 Answer
To use the number line to represent \(\frac{1}{6}\) x 3 = \(\frac{1}{2}\), we first need to divide the number line into sixths since the first factor of the multiplication sentence is \(\frac{1}{6}\).
Since the second factor is 3, we need to mark off three \(\frac{1}{6}\) parts, as shown below by the red ovals.
Note that the last oval ends at \(\frac{3}{6}\) = \(\frac{1}{2}\) which is why the product \(\frac{1}{6}\) × 3 is equal to \(\frac{1}{2}\).
To write an equivalent division sentence for the number line, we need to look at the total width of the ovals to get the dividend, the number of ovals to get the divisor, and the width of each oval to get the quotient.
The ovals have a total width of \(\frac{1}{2}\) so the dividend is \(\frac{1}{2}\). The width of each oval is \(\frac{1}{6}\) so the divisor is \(\frac{1}{6}\). There are 3 ovals so the quotient is 3.
An equivalent division sentence is then \(\frac{1}{2} \div \frac{1}{6}=3\)
Result
To represent \(\frac{1}{6}\) x 3 = \(\frac{1}{2}\) on the number line, draw 3 ovals with a width of \(\frac{1}{6}\), starting from 0. The equivalent division statement for the number line is \(\frac{1}{2}\) ÷ \(\frac{1}{6}\) = 3.
The dividend is represented in the number line as the total width of the ovals, the divisor is the width of each oval, and the quotient is the number of ovals.
Page 39 Exercise 2 Answer
To find \(\frac{1}{4} \div \frac{3}{8}\) using an area model, start by drawing two rectangles. Divide the first rectangle into fourths and shade 1 of them and divide the second rectangle into eights and shade 3 of them, as shown below in Step 1.
Next, we need to divide the two rectangles so they have the same number of parts. Divide each fourth into halves so the top rectangle will have 8 parts like the bottom rectangle, as shown below in Step 2.
Since \(\frac{1}{4}\) is divided into 2 equal parts while \(\frac{3}{8}\) is divided into 3 equal parts, then \(\frac{1}{4} \div \frac{3}{8}=\frac{2}{3}\).
Result
\(\frac{2}{3}\)
Page 40 Exercise 1 Answer
To divide a fraction by a fraction,
rewrite the division problem as a multiplication problem, that is, multiply the dividend by the reciprocal of the divisor.
For example,
\(\frac{3}{4} \div \frac{1}{5}=\frac{3}{4} \times \frac{5}{1}\)
Result
To divide a fraction by a fraction, rewrite the division problem as a multiplication problem, that is, multiply the dividend by the reciprocal of the divisor.
Page 40 Exercise 2 Answer
Division is not commutative that is a dividend and a dividor can’t switch places. When rewriting the division problem as a mulitplication problem multiply the dividend by the reciprocal of a divisor.
In Corey’s problem \(\frac{2}{5}\) is the dividend and \(\frac{8}{5}\) is the divisor. So the correct answer would be multiplying \(\frac{2}{5}\) by the reciprocal of \(\frac{8}{5}\) which is \(\frac{5}{8}\).
\(\frac{2}{5} \div \frac{8}{5}=\frac{2}{5} \times \frac{5}{8}\)
Corey’s mistake is that he multiplied the divisor by a reciprocal of a dividend.
Result
Corey’s mistake is that he multiplied the divisor by a reciprocal of a dividend. The correct expression
\(\frac{2}{5} \div \frac{8}{5}=\frac{2}{5} \times \frac{5}{8}\)
Page 40 Exercise 3 Answer
The quotient of \(\frac{3}{5}\) ÷ \(\frac{6}{7}\) is greater than \(\frac{3}{5}\) since the divisor is less than 1.
Result
The quotient is greater than \(\frac{3}{5}\).
Page 40 Exercise 4 Answer
To divide a fraction by a whole number, we must first rewrite the whole number as a fraction and then rewrite the division as multiplying by the reciprocal of the divisor. For example:
\(\frac{3}{7} \div 5=\frac{3}{7} \div \frac{5}{1}=\frac{3}{7} \times \frac{1}{5}\)
Also, when dividing a fraction by a whole number, the quotient is always smaller than the dividend.
To divide a fraction by a fraction, we go directly to rewriting the division as multiplying by the reciprocal of the divisor. For example:
\(\frac{3}{7} \div \frac{2}{3}=\frac{3}{7} \times \frac{3}{2}\)
When dividing a fraction by a fraction, the quotient is smaller than the dividend if the divisor is a fraction greater than 1 and the quotient is bigger than the dividend if the divisor is a fraction smaller than 1.
Result
To divide a fraction by a whole number, we must first rewrite the whole number as a fraction and then rewrite the division as multiplying. The quotient is always smaller than the dividend.
To divide a fraction by a fraction, we go directly to rewriting the division as multiplying. The quotient is smaller than the dividend if the divisor is a fraction greater than 1 and the quotient is bigger than the dividend if the divisor is a fraction smaller than 1.
Page 40 Exercise 5 Answer
To write a division sentence to represent the model, we need to determine the total width of the ovals, the number of ovals, and the width of each oval.
The dividend is the total width of the ovals, which is \(\frac{4}{5}\).
The divisor is the width of each oval, which is \(\frac{1}{5}\).
The quotient is the number of ovals, which is 4.
The division sentence is then \(\frac{4}{5} \div \frac{1}{5}=4\)
Result
\(\frac{4}{5} \div \frac{1}{5}=4\)
Page 40 Exercise 6 Answer
To write a division sentence to represent the model, we need to see what fraction of each rectangle is shown and how many equal parts are in each fraction.
The top rectangle is labeled as \(\frac{1}{2}\) so the dividend is \(\frac{1}{2}\).
The bottom rectangle is labeled as \(\frac{2}{3}\) so the divisor is \(\frac{2}{3}\)
\(\frac{1}{2}\) is divided into 3 equal parts while \(\frac{2}{3}\) is divided into 4 equal parts. The quotient is then \(\frac{3}{4}\).
Therefore, the division sentence is \(\frac{1}{2} \div \frac{2}{3}=\frac{3}{4}\)
Result
\(\frac{1}{2} \div \frac{2}{3}=\frac{3}{4}\)
Page 40 Exercise 7 Answer
To write a division sentence for the model, we need to determine what fraction of the whole rectangle is shaded, how many parts the rectangle is divided into, and how many parts are shaded.
The rectangle is initially divided into 4 parts and 3 parts are shaded to represent \(\frac{3}{4}\) so the dividend is \(\frac{3}{4}\).
The rectangle is then divided into eighths so the divisor is \(\frac{1}{8}\).
There are 6 eighths shaded so the quotient is 6.
The division sentence is then \(\frac{3}{4} \div \frac{1}{8}=6\).
Result
\(\frac{3}{4} \div \frac{1}{8}=6\)
Page 40 Exercise 8 Answer
\(\frac{3}{4} \div \frac{2}{3}\) = \(\frac{3}{4}\) x \(\frac{3}{2}\) (Rewrite as a multiplication problem)
= \(\frac{3 \times 3}{4 \times 2}\)
= \(\frac{9}{8}\) (Multiply the fractions)
= \(\frac{8}{8}+\frac{1}{8}\)
= \(1 \frac{1}{8}\)
Result
= \(1 \frac{1}{8}\)
Page 40 Exercise 9 Answer
\(\frac{3}{12} \div \frac{1}{8}=\frac{1}{4} \div \frac{1}{8}\) (Rewrite \(\frac{3}{12}\) as \(\frac{1}{4}\))
= \(\frac{1}{4}\) x \(\frac{8}{1}\) (Rewrite as a multiplication problem)
= \(\frac{8}{4}\) (Multiply the fractions)
= 2
Result
2
Page 40 Exercise 10 Answer
\(\frac{1}{2} \div \frac{4}{5}\) = \(\frac{1}{2}\) x \(\frac{5}{4}\) (Rewrite as a multiplication problem)
= \(\frac{5}{8}\) (Multiply the fractions)
Result
\(\frac{5}{8}\)
Page 40 Exercise 11 Answer
\(\frac{7}{10} \div \frac{2}{5}\) = \(\frac{7}{10}\) x \(\frac{5}{2}\) (Rewrite as a multiplication problem)
= \(\frac{35}{20}\) (Multiply the fractions)
= \(\frac{7}{4}\)
Result
\(\frac{7}{4}\)
Page 41 Exercise 12 Answer
When representing a division sentence using a number line, the number of ovals is the quotient.
From the given number line, there are 4 ovals so \(\frac{1}{3} \div \frac{1}{2}=4\).
Result
4
Page 41 Exercise 13 Answer
When using an area model to represent a division sentence, the number of shaded parts is the quotient.
From the given diagram, the rectangle has 4 shaded parts so \(\frac{2}{5} \div \frac{1}{10}=4\).
Result
4
Page 41 Exercise 14 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{2}{3} \div \frac{1}{3}=\frac{2}{3} \times \frac{3}{1}=\frac{2 \times 3}{3 \times 1}=\frac{6}{3}=2\)
Result
2
Page 41 Exercise 15 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{1}{2} \div \frac{1}{16}=\frac{1}{2} \times \frac{16}{1}=\frac{1 \times 16}{2 \times 1}=\frac{16}{2}=8\)
Result
8
Page 41 Exercise 16 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{1}{4} \div \frac{1}{12}=\frac{1}{4} \times \frac{12}{1}=\frac{1 \times 12}{4 \times 1}=\frac{12}{4}=3\)
Result
3
Page 41 Exercise 17 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{6}{7} \div \frac{3}{7}=\frac{6}{7} \times \frac{7}{3}=\frac{6 \times 7}{7 \times 3}=\frac{42}{21}=2\)
Result
2
Page 41 Exercise 18 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{5}{14} \div \frac{4}{7}=\frac{5}{14} \times \frac{7}{4}=\frac{5 \times 7}{14 \times 4}=\frac{35}{56}=\frac{5}{8}\)
Result
\(\frac{5}{8}\)
Page 41 Exercise 19 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{5}{8} \div \frac{1}{2}=\frac{5}{8} \times \frac{2}{1}=\frac{5 \times 2}{8 \times 1}=\frac{10}{8}=\frac{5}{4}=\frac{4}{4}+\frac{1}{4}=1 \frac{1}{4}\)
Result
\(1 \frac{1}{4}\)
Page 41 Exercise 20 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{7}{12} \div \frac{3}{4}=\frac{7}{12} \times \frac{4}{3}=\frac{7 \times 4}{12 \times 3}=\frac{28}{36}=\frac{7}{9}\)
Result
\(\frac{7}{9}\)
Page 41 Exercise 21 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{2}{7} \div \frac{1}{2}=\frac{2}{7} \times \frac{2}{1}=\frac{2 \times 2}{7 \times 1}=\frac{4}{7}\)
Result
\(\frac{4}{7}\)
Page 41 Exercise 22 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{4}{9} \div \frac{2}{3}=\frac{4}{9} \times \frac{3}{2}=\frac{4 \times 3}{9 \times 2}=\frac{12}{18}=\frac{2}{3}\)
Result
\(\frac{2}{3}\)
Page 41 Exercise 23 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{7}{12} \div \frac{1}{8}=\frac{7}{12} \times \frac{8}{1}=\frac{7 \times 8}{12 \times 1}=\frac{56}{12}=\frac{14}{3}=\frac{12}{3}+\frac{2}{3}=4 \frac{2}{3}\)
Result
\(4 \frac{2}{3}\)
Page 41 Exercise 24 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{3}{10} \div \frac{3}{5}=\frac{3}{10} \times \frac{5}{3}=\frac{3 \times 5}{10 \times 3}=\frac{15}{30}=\frac{1}{2}\)
Result
\(\frac{1}{2}\)
Page 41 Exercise 25 Answer
To divide a fraction by a fraction, rewrite the division equation as a multiplication equation. To divide by a fraction, a multiply by the reciprocal of the divisor.
\(\frac{2}{5} \div \frac{1}{8}=\frac{2}{5} \times \frac{8}{1}=\frac{2 \times 8}{5 \times 1}=\frac{16}{5}=\frac{15}{5}+\frac{1}{5}=3 \frac{1}{5}\)
Result
\(3 \frac{1}{5}\)
Page 41 Exercise 26 Answer
We need to determine how many \(\frac{1}{3}\) pound bags can be filled with \(\frac{12}{15}\) pounds of granola. To find the number of bags, we must then divide \(\frac{12}{15}\) and \(\frac{1}{3}\).
To divide by a fraction, rewrite the division as multiplying by the reciprocal of the divisor:
\(\frac{12}{15} \div \frac{1}{3}=\frac{12}{15} \times \frac{3}{1}=\frac{12 \times 3}{15 \times 1}=\frac{36}{15}=\frac{12}{5}=2 \frac{2}{5}\)
We can then completely fill 2 bags and can fill \(\frac{2}{5}\) of another bag.
To find how much granola is left, we need to find how much granola is \(\frac{2}{5}\) of a bag. Since each bag is \(\frac{1}{3}\) pound, we can multiply \(\frac{2}{5}\) and \(\frac{1}{3}\):
\(\frac{2}{5} \times \frac{1}{3}=\frac{2 \times 1}{5 \times 3}=\frac{2}{15}\)
The amount of leftover granola is then \(\frac{2}{15}\) pound.
Result
2 bags can be filled and \(\frac{2}{15}\) pound is left over.
Page 41 Exercise 27 Answer
To divide by a fraction, rewrite the division as multiplying by the reciprocal of the divisor:
\(\frac{3}{4} \div \frac{2}{3}=\frac{3}{4} \times \frac{3}{2}=\frac{3 \times 3}{4 \times 2}=\frac{9}{8}=1 \frac{1}{8}\)
To use a model to find the quotient, we can start by drawing two rectangles. Divide the first rectangle into fourths and shade three of them to represent \(\frac{3}{4}\). Divide the second rectangle into thirds and shade two of them to represent \(\frac{2}{3}\).
Next, divide each rectangle into the same number of parts. Since 4 × 3 = 12, we can divide each rectangle into 12 equal parts.
\(\frac{3}{4}\) is divided into 9 equal parts while \(\frac{2}{3}\) is divided into 8 equal parts. Therefore, \(\frac{3}{4} \div \frac{2}{3}=\frac{9}{8}=1 \frac{1}{8}\)
Result
\(1 \frac{1}{8}\)
Page 41 Exercise 28 Answer
Using the formula A = l x w
A = \(\frac{1}{6}\) W = \(\frac{2}{3}\)
\(A=l \times w \rightarrow \frac{1}{6}=l \times \frac{2}{3} \rightarrow l=\frac{1}{6} \div \frac{2}{3}=\frac{1}{6} \times \frac{3}{2}=\frac{1 \times 3}{6 \times 2}=\frac{3}{12}=\frac{1}{4}\)
Thus, the length of the painting is \(\frac{1}{4}\) yard.
Result
l = \(\frac{1}{4}\)
Page 41 Exercise 29 Answer
Solve the equation \(\frac{13}{16} \div \frac{1}{6}=n\) or n by rewriting the division as multiplying by the reciprocal of the divisor:
\(n=\frac{13}{16} \div \frac{1}{6}=\frac{13}{16} \times \frac{6}{1}=\frac{13 \times 6}{16 \times 1}=\frac{78}{16}=\frac{39}{8}=\frac{32}{8}+\frac{7}{8}=4 \frac{7}{8}\)
Result
n = \(4 \frac{7}{8}\)
Page 42 Exercise 30a Answer
It is given that the cafeteria has \(\frac{2}{3}\) pound of coffee and uses \(\frac{1}{6}\) pound to fill the dispenser. To find how many times the dispenser can be filled, we then need to find the quotient \(\frac{2}{3} \div \frac{1}{6}\).
To find the quotient using a model, start by representing the dividend. Divide the rectangle into 3 parts and shade 2 of them to represent the dividend \(\frac{2}{3}\).
Next, divide the rectangle into sixths to represent the divisor \(\frac{1}{6}\).
Since 4 parts of the rectangle are shaded, then the quotient is 4 so \(\frac{2}{3} \div \frac{1}{6}=4\)
The cafeteria can then fill 4 dispensers.
Result
4 dispensers
Page 42 Exercise 30b Answer
From part (a), we know the division sentence is \(\frac{2}{3} \div \frac{1}{6}=4\)
Since the quotient is 4, the cafeteria can fill 4 dispensers.
Result
\(\frac{2}{3} \div \frac{1}{6}=4\) 4 dispensers
Page 42 Exercise 31a Answer
We know that the truck can haul \(\frac{2}{3}\) ton and is currently hauling \(\frac{1}{2}\) ton.
To find how much of a full load the truck is hauling, we must then find the quotient \(\frac{1}{2} \div \frac{2}{3}\)
To find the quotient using the model, start by dividing the first rectangle into halves and shade 1 half to represent the dividend \(\frac{1}{2}\). Shade the second rectangle into thirds and shade 2 thirds to represent the divisor \(\frac{2}{3}\).
Next, we need to divide the two rectangles into the same number of parts. Since 2 × 3 = 6, we can divide both rectangles into 6 equal parts.
\(\frac{1}{2}\) is divided into 3 equal parts while \(\frac{2}{3}\) is divided into 4 equal parts. The quotient is then \(\frac{1}{2} \div \frac{2}{3}=\frac{3}{4}\).
The truck is then currently hauling \(\frac{3}{4}\) of a full load.
Result
\(\frac{3}{4}\) of a full load
Page 42 Exercise 31b Answer
From part (a), we know the division sentence is \(\frac{1}{2} \div \frac{2}{3}=\frac{3}{4}\)
Since the quotient is \(\frac{3}{4}\), then the truck is hauling \(\frac{3}{4}\) of a full load
Result
\(\frac{1}{2} \div \frac{2}{3}=\frac{3}{4}\)
\(\frac{3}{4}\) of a full load
Page 42 Exercise 32 Answer
To answer the question find the quotient \(\frac{5}{8} \div \frac{1}{4}\), since the piece of metal is \(\frac{5}{8}\) inch long and is cut into \(\frac{1}{4}\) inch pieces.
\(\frac{5}{8} \div \frac{1}{4}\) = \(\frac{5}{8}\) x \(\frac{4}{1}\) (Rewrite as a multiplication problem)
= \(\frac{5 \times 4}{8 \times 1}\)
= \(\frac{20}{8}\) (Multiply the fractions)
= \(\frac{5}{2}\)
= \(\frac{4}{2}+\frac{1}{2}\)
= \(2 \frac{1}{2}\) (Rewrite as a mixed number)
The number of \(\frac{1}{4}\) inch long pieces must be a whole number. A piece of metal \(\frac{5}{8}\) inch long can then be divided into 2 pieces that are each \(\frac{1}{4}\) inch long.
Result
2 pieces that are \(\frac{1}{4}\) inch long
Page 42 Exercise 33 Answer
Margaret has a rope \(\frac{5}{8}\) inch long. She wants to cut it into smaller pieces, each the same length of \(\frac{2}{5}\) inch. How many \(\frac{2}{5}\) inch pieces of rope will she have and will there be any left over rope?
\(\frac{5}{8} \div \frac{2}{5}\) = \(\frac{5}{8}\) x \(\frac{5}{2}\) (Rewrite as a multiplication problem)
= \(\frac{5 \times 5}{8 \times 2}\)
= \(\frac{25}{16}\) (Multiply the fractions)
= \(\frac{16}{16}+\frac{9}{16}\)
= \(1 \frac{9}{16}\)
The quotient is a mixed number but the number of \(\frac{2}{5}\) inch pieces must be a whole number. This means she will have 1 piece that is \(\frac{2}{5}\) inch and will have left over rope.
Result
Possible solution: Margaret has a rope \(\frac{5}{8}\) inch long. She wants to cut it into smaller pieces, each the same length of \(\frac{2}{5}\) inch. How many \(\frac{2}{5}\) inch pieces of rope will she have and will there be any left over rope?
She will have 1 piece that is \(\frac{2}{5}\) inch and will have left over rope.
Page 42 Exercise 34 Answer
The colored in rectangle represents \(\frac{2}{3}\). Since each of the smallest rectangles in the model represent \(\frac{1}{9}\), the model represents the quotient \(\frac{2}{3} \div \frac{1}{9}\)
This answer is marked A, that is \(\frac{2}{3} \div \frac{1}{9}\) = 6.
Result
A \(\frac{2}{3} \div \frac{1}{9}\) = 6
