Chapter 1 Use Positive Rational Numbers
Section 1.4: Understand Division With Fractions
Page 31 Exercise 1a Answer
Each point represents the position of each runner after his turn:
Result
Plot points which represent the position of each runner after their turn.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 31 Exercise 1b Answer
Step 1
We can find each point (points represent distances of each runner from the start) by multiplying \(\frac{2}{5}\) by the turn number of each runner.
So, the points which represent each runner after his turn can be obtained by the following procedure:
\(\frac{2}{5} \times 1=\frac{2}{5}\) \(\frac{2}{5} \times 2=\frac{4}{5}\) \(\frac{2}{5} \times 3=\frac{6}{5}\) \(\frac{2}{5} \times 4=\frac{8}{5}\) \(\frac{2}{5} \times 5=2\) \(\frac{2}{5} \times 6=\frac{12}{5}\) \(\frac{2}{5} \times 7=\frac{14}{5}\) \(\frac{2}{5} \times 8=\frac{16}{5}\) \(\frac{2}{5} \times 9=\frac{18}{5}\) \(\frac{2}{5} \times 10=4\)Result
Multiply \(\frac{2}{5}\) by the turn number of each runner.
Page 31 Exercise 1 Answer
The line would have values between 0 and 5, and the distance between adjacent points is \(\frac{1}{2}\). Each point can be found by multiplying \(\frac{1}{2}\) by turn number of each runner.
Result
Each point can be found by multiplying \(\frac{1}{2}\) by turn number of each runner.
Page 32 Exercise 1 Answer
We need to determine how many \(\frac{2}{3}\) foot long pieces can be made from a 6-foot long board. We must find \(6 \div \frac{2}{3}\) using the given number line.
Label the given number line from 0 to 6 and then divide each segment into thirds.
Next, divide the number line into \(\frac{2}{3}\) foot parts, as shown by the red arcs below.
Since the number line from 0 to 6 can be divided into 9 parts, then \(6 \div \frac{2}{3}=9\). Therefore, 9 pieces can be cut from the board.
When dividing by a number that is less than 1, the quotient is always greater than the dividend. Since the divisor is \(\frac{2}{3}\), which is less than 1, then the quotient must be greater than the dividend of 6.
Result
9 pieces
Since the divisor is \(\frac{2}{3}\), which is less than 1, then the quotient must be greater than the dividend of 6.
Page 33 Exercise 2 Answer
To make a diagram to find \(\frac{2}{3} \div 4\), start by drawing a rectangle and dividing it into thirds. Shade 2 of the thirds, as shown in step 1 below.
The divide the rectangle into 4 equal parts, as shown in step 2 below.
Each part is \(\frac{1}{12}\) of the whole and each fourth has two shaded parts. We then get:
\(\frac{2}{3} \div 4=\frac{2}{12}=\frac{1}{6}\)
Result
\(\frac{1}{6}\)Page 33 Exercise 3 Answer
To divide a whole number rewrite the problem as a multiplication problem using the reciprocal of the divisor.
\(8 \div \frac{3}{4}=8 \times \frac{4}{3}=\frac{8}{1} \times \frac{4}{3}=\frac{8 \times 4}{1 \times 3}=\frac{32}{3}=\frac{30}{3}+\frac{2}{3}=10 \frac{2}{3}\)Result
\(10 \frac{2}{3}\)Page 34 Exercise 4 Answer
When the divisor is a fraction less than 1, the quotient is greater than the dividend.
Page 34 Exercise 5 Answer
When the divisor is a fraction less than 1, the quotient is greater than the dividend.
Page 34 Exercise 6 Answer
To find the division equation represented by the diagram, we need to look to see how many parts the rectangle is divided up into and how many parts are shaded.
The rectangle is divided into 4 columns and 3 of the columns are shaded so the dividend is \(\frac{3}{4}\).
The rectangle is divided into 2 rows so the divisor is 2.
Since the rectangle is divided up into 8 parts and 3 of those parts are circled, then the quotient is \(\frac{3}{8}\).
The division equation is then \(\frac{3}{4} \div 2=\frac{3}{8}\)
Result
\(\frac{3}{4} \div 2=\frac{3}{8}\)Page 34 Exercise 7 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).
Result
\(\frac{5}{3}\)Page 34 Exercise 8 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{1}{6}\) is \(\frac{6}{1}\), which is 6.
Result
6
Page 34 Exercise 9 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of 9, which can be written as \(\frac{9}{1}\) is \(\frac{1}{9}\).
Result
\(\frac{1}{9}\)Page 34 Exercise 10 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{7}{4}\) is \(\frac{4}{7}\).
Result
\(\frac{4}{7}\)Page 34 Exercise 11 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{5}{8}\) is \(\frac{8}{5}\).
Result
\(\frac{8}{5}\)Page 34 Exercise 12 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of 16, which can be written as \(\frac{16}{1}\) is \(\frac{1}{16}\).
Result
\(\frac{1}{16}\)Page 34 Exercise 13 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{7}{12}\) is \(\frac{12}{7}\).
Result
\(\frac{12}{7}\)Page 34 Exercise 14 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{11}{5}\) is \(\frac{5}{11}\).
Result
\(\frac{5}{11}\)Page 34 Exercise 15 Answer
Divide 6 by \(\frac{2}{3}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(6 \div \frac{2}{3}=\frac{6}{1} \div \frac{2}{3}=\frac{6}{1} \times \frac{3}{2}=\frac{6 \times 3}{1 \times 2}=\frac{18}{2}=9\)Result
9
Page 34 Exercise 16 Answer
Divide 6 by \(\frac{3}{8}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(12 \div \frac{3}{8}=\frac{12}{1} \div \frac{3}{8}=\frac{12}{1} \times \frac{8}{3}=\frac{12 \times 8}{1 \times 3}=\frac{96}{3}=32\)Result
32
Page 34 Exercise 17 Answer
Divide \(\frac{1}{4}\) by 3.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{1}{4} \div 3=\frac{1}{4} \div \frac{3}{1}=\frac{1}{4} \times \frac{1}{3}=\frac{1 \times 1}{4 \times 3}=\frac{1}{12}\)Result
\(\frac{1}{12}\)Page 34 Exercise 18 Answer
Divide \(\frac{2}{5}\) by 3.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{2}{5} \div 2=\frac{2}{5} \div \frac{2}{1}=\frac{2}{5} \times \frac{1}{2}=\frac{2 \times 1}{5 \times 2}=\frac{2}{10}=\frac{1}{5}\)Result
\(\frac{1}{5}\)Page 34 Exercise 19 Answer
Divide 2 by \(\frac{1}{2}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(2 \div \frac{1}{2}=\frac{2}{1} \div \frac{1}{2}=\frac{2}{1} \times \frac{2}{1}=\frac{2 \times 2}{1 \times 1}=\frac{4}{1}=4\)Result
4
Page 34 Exercise 20 Answer
Divide 3 by \(\frac{1}{4}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(3 \div \frac{1}{4}=\frac{3}{1} \div \frac{1}{4}=\frac{3}{1} \times \frac{4}{1}=\frac{3 \times 4}{1 \times 1}=\frac{12}{1}=12\)Result
12
Page 34 Exercise 21 Answer
Divide 9 by \(\frac{3}{5}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(9 \div \frac{3}{5}=\frac{9}{1} \div \frac{3}{5}=\frac{9}{1} \times \frac{5}{3}=\frac{9 \times 5}{1 \times 3}=\frac{45}{3}=15\)Result
15
Page 34 Exercise 22 Answer
Divide 5 by \(\frac{2}{7}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(5 \div \frac{2}{7}=\frac{5}{1} \div \frac{2}{7}=\frac{5}{1} \times \frac{7}{2}=\frac{5 \times 7}{1 \times 2}=\frac{35}{2}\)Result
\(\frac{35}{2}\)Page 35 Exercise 23 Answer
When using a number line to represent division by a fraction, the width of each oval is the divisor.
From the diagram, we can see there are two ovals between each pair of consecutive whole numbers. This means that each oval in the diagram has a width of \(\frac{1}{2}\) so the divisor must then be \(\frac{1}{2}\).
The division sentence is then completed as: \(6 \div \frac{1}{2}=12\)
Result
\(6 \div \frac{1}{2}=12\)Page 35 Exercise 24 Answer
The number of columns in the diagram tells us the dividend and the number of rows tells us the divisor.
From the diagram, the rectangle is dividend into 3 rows so the divisor must be 3.
The completed division sentence is then: \(\frac{2}{3} \div 3=\frac{2}{9} .\)
Result
\(\frac{2}{3} \div 3=\frac{2}{9}\)Page 35 Exercise 25 Answer
Divide \(\frac{3}{5}\) by 3.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{3}{5} \div 3=\frac{3}{5} \div \frac{3}{1}=\frac{3}{5} \times \frac{1}{3}=\frac{3 \times 1}{5 \times 3}=\frac{3}{15}=\frac{1}{5}\)Result
\(\frac{1}{5}\)Page 35 Exercise 26 Answer
Divide 2 by \(\frac{2}{5}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(2 \div \frac{2}{5}=\frac{2}{1} \div \frac{2}{5}=\frac{2}{1} \times \frac{5}{2}=\frac{2 \times 5}{1 \times 2}=\frac{10}{2}=5\)Result
5
Page 35 Exercise 27 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{3}{10}\) is \(\frac{10}{3}\).
Result
\(\frac{10}{3}\)Page 35 Exercise 28 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of 6, which can be written as \(\frac{6}{1}\) is \(\frac{1}{6}\).
Result
\(\frac{1}{6}\)Page 35 Exercise 29 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of \(\frac{1}{15}\) is \(\frac{15}{1}\), which can be written as 15.
Result
15
Page 35 Exercise 30 Answer
Two numbers whose product is one are called reciprocals of each other. If a nonzero number is named as a fraction \(\frac{a}{b}\), then its reciprocal is \(\frac{b}{a}\).
The reciprocal of 3, which can be written as \(\frac{3}{1}\) is \(\frac{1}{3}\).
Result
\(\frac{1}{3}\)Page 35 Exercise 31 Answer
Divide 36 by \(\frac{3}{4}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(36 \div \frac{3}{4}=\frac{36}{1} \div \frac{3}{4}=\frac{36}{1} \times \frac{4}{3}=\frac{36 \times 4}{1 \times 3}=\frac{144}{3}=48\)Result
48
Page 35 Exercise 32 Answer
Divide 2 by \(\frac{3}{8}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(2 \div \frac{3}{8}=\frac{2}{1} \div \frac{3}{8}=\frac{2}{1} \times \frac{8}{3}=\frac{2 \times 8}{1 \times 3}=\frac{16}{3}\)Result
\(\frac{16}{3}\)Page 35 Exercise 33 Answer
Divide 18 by \(\frac{2}{3}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(18 \div \frac{2}{3}=\frac{18}{1} \div \frac{2}{3}=\frac{18}{1} \times \frac{3}{2}=\frac{18 \times 3}{1 \times 2}=\frac{54}{2}=27\)Result
27
Page 35 Exercise 34 Answer
Divide 9 by \(\frac{4}{5}\).
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(9 \div \frac{4}{5}=\frac{9}{1} \div \frac{4}{5}=\frac{9}{1} \times \frac{5}{4}=\frac{9 \times 5}{1 \times 4}=\frac{45}{4}\)Result
\(\frac{45}{4}\)Page 35 Exercise 35 Answer
Divide \(\frac{1}{6}\) by 2.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{1}{6} \div 2=\frac{1}{6} \div \frac{2}{1}=\frac{1}{6} \times \frac{1}{2}=\frac{1 \times 1}{6 \times 2}=\frac{1}{12}\)Result
\(\frac{1}{12}\)Page 35 Exercise 36 Answer
Divide \(\frac{2}{3}\) by 3.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{2}{3} \div 3=\frac{2}{3} \div \frac{3}{1}=\frac{2}{3} \times \frac{1}{3}=\frac{2 \times 1}{3 \times 3}=\frac{2}{9}\)Result
\(\frac{2}{9}\)Page 35 Exercise 37 Answer
Divide \(\frac{3}{5}\) by 2.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{3}{5} \div 2=\frac{3}{5} \div \frac{2}{1}=\frac{3}{5} \times \frac{1}{2}=\frac{3 \times 1}{5 \times 2}=\frac{3}{10}\)Result
\(\frac{3}{10}\)Page 35 Exercise 38 Answer
Divide \(\frac{1}{4}\) by 4.
To divide a whole number by a fraction, first write the whole number as a fraction and than multiply the fraction by the reciprocal of the divisor.
\(\frac{1}{4} \div 4=\frac{1}{4} \div \frac{4}{1}=\frac{1}{4} \times \frac{1}{4}=\frac{1 \times 1}{4 \times 4}=\frac{1}{16}\)Result
\(\frac{1}{16}\)Page 35 Exercise 39 Answer
A worker is pouring 3 quart of liquid into \(\frac{3}{8}\) quart containers.
\(3 \div \frac{3}{8}=\frac{3}{1} \div \frac{3}{8}=\frac{3}{1} \times \frac{8}{3}=\frac{3 \times 8}{1 \times 3}=\frac{24}{3}=8\)Result
A worker can fill 8 containers.
Page 36 Exercise 40 Answer
A snail can move 120 ft in \(\frac{3}{4}\) h. A tortoise can move 600 ft in \(\frac{2}{3}\) h. A sloth can move 250 ft in \(\frac{5}{8}\) h.
The animal which moves the fastest is the one that moves more feet in a shorter period of time.
Comparing the snail and tortoise, \(\frac{2}{3}\) of an hour is less than \(\frac{3}{4}\) and 120 ft is less than 600 ft, so a tortoise moves faster than a snail since it moves more feet in a shorter period of time.
Comparing the sloth and tortoise, \(\frac{5}{8}\) is close to \(\frac{2}{3}\) but 250 ft is less than 600 ft, so a tortoise moves faster than a sloth since it moves more feet in about the same amount of time.
Result
The animal which moves the fastest is the one that moves more feet in a shorter period of time so the tortoise moves the fastest.
Page 36 Exercise 41 Answer
Since the quotient 250 ÷ \(\frac{5}{8}\) tells us about how far a sloth moves in 1 hour and 90 minutes = \(1 \frac{1}{2}\) hour, we can multiply the quotient by \(1 \frac{1}{2}\) to find how far teh sloth can go in 90 minutes.
First, let’s find the quotient. To divide by a fraction, rewrite the expressions as multiplying by the reciprocal of the divisor and then multiply:
\(250 \div \frac{5}{8}=\frac{250}{1} \times \frac{8}{5}=\frac{250 \times 8}{1 \times 5}=\frac{2000}{5}=400\)The sloth will then move about 400 feet in 1 hour. Multiplying this distance by \(1 \frac{1}{2}\) gives:
\(400 \times 1 \frac{1}{2}=\frac{400}{1} \times \frac{3}{2}=\frac{400 \times 3}{1 \times 2}=\frac{1200}{2}=600\)Therefore, the sloth can move 600 ft in 90 minutes.
Result
600 ft
Page 36 Exercise 42 Answer
A tortoise can move 600 ft in \(\frac{2}{3}\)h.
The quotient 600 ÷ \(\frac{2}{3}\) tells about how far a tortoise may move in one hour.
\(600 \div \frac{2}{3}=\frac{600}{1} \div \frac{2}{3}=\frac{600}{1} \times \frac{3}{2}=\frac{600 \times 3}{1 \times 2}=\frac{1800}{2}=900\)Result
A tortoise may move 900 ft in one hour.
Page 36 Exercise 43 Answer
A snail can move 120 ft in \(\frac{3}{4}\) h, so the quotient 120 ÷ \(\frac{3}{4}\) how far a snail can move in one hour.
\(120 \div \frac{3}{4}=\frac{120}{1} \div \frac{3}{4}=\frac{120}{1} \times \frac{4}{3}=\frac{120 \times 4}{1 \times 3}=\frac{480}{3}=160\)Result
A snail can move 160 ft in one hour.
Page 36 Exercise 44 Answer
\(\frac{3}{4}\) gallon of juice is being divided equally into 5 pitchers so we need to represent the quotient \(\frac{3}{4}\) ÷ 5 using the rectangle.
The rectangle represents 1 whole gallon so first we need to divide it into 4 equal columns and shade 3 of them to represent the \(\frac{3}{4}\) gallons.
Next, we need to divide the rectangle into 5 rows to represent the 5 pitchers.
The whole rectangle is divided into 20 parts in all. Each row has 3 parts that are shaded so the quotient is \(\frac{3}{20}\). The division equation is then \(\frac{3}{4} \div 5=\frac{3}{20}\) and each pitcher has \(\frac{3}{20}\) gallon of juice.
Result
\(\frac{3}{20}\) gallon
\(\frac{3}{4} \div 5=\frac{3}{20}\)Page 36 Exercise 45 Answer
Five equation are give and we have to select all those which are true.
To divide a number with a fraction is the same as multiply the number with the reciprocal of that fraction. Thus, the following equations are true.
\(14 \div \frac{7}{10}=14 \times \frac{10}{7}\) \(10 \div \frac{3}{5}=10 \times \frac{5}{3}\) \(20 \div 4=20 \times \frac{1}{4}\)Division is not commutative, which means it is not the same, for example, to divide 15 by 3, and to divide 3 by 15.
\(16 \div \frac{4}{5} \stackrel{?}{=} \frac{1}{16} \times \frac{4}{5}\) \(16 \div \frac{4}{5}=\frac{16}{1} \times \frac{5}{4}=\frac{16 \times 5}{1 \times 4}=\frac{80}{4}=20\) \(\frac{1}{16} \times \frac{4}{5}=\frac{1 \times 4}{16 \times 5}=\frac{4}{80}=\frac{1}{20}\) \(20 \neq \frac{1}{20}\) \(16 \div \frac{4}{5} \neq \frac{1}{16} \times \frac{4}{5}\)The same logic is used to show that \(12 \div \frac{2}{3} \neq \frac{1}{12} \times \frac{2}{3}\)
Result
\(14 \div \frac{7}{10}=14 \times \frac{10}{7}, 10 \div \frac{3}{5}=10 \times \frac{5}{3} \text {, and } 20 \div 4=20 \times \frac{1}{4}\)Page 36 Exercise 46 Answer
Five equation are give and we have to select all those which are true.
To divide a number with a fraction is the same as multiply the number with the reciprocal of that fraction. Thus, the following equations are true.
\(\frac{1}{3} \div 3=\frac{1}{3} \div \frac{3}{1}=\frac{1}{3} \times \frac{1}{3}\) \(\frac{4}{5} \div 5=\frac{4}{5} \div \frac{5}{1}=\frac{4}{5} \times \frac{1}{5}\) \(\frac{2}{3} \div 6=\frac{2}{3} \div \frac{6}{1}=\frac{2}{3} \times \frac{1}{6}\)Every whole number can be written as a fraction with that number as the numerator and one as the denominator.
8 written as a fraction is \(\frac{8}{1}\), not \(\frac{1}{8}\).
Thus, the equation \(\frac{7}{8} \div 8=\frac{7}{8} \div \frac{1}{8}=\frac{7}{8} \times \frac{8}{1}\) is not true.
The same logic is used to show that the equation \(\frac{4}{9} \div 4=\frac{4}{9} \div \frac{1}{4}=\frac{4}{9} \times \frac{4}{1}\) is also not true.
Result
\(\frac{1}{3} \div 3=\frac{1}{3} \div \frac{3}{1}=\frac{1}{3} \times \frac{1}{3}\) \(\frac{4}{5} \div 5=\frac{4}{5} \div \frac{5}{1}=\frac{4}{5} \times \frac{1}{5}\) \(\frac{2}{3} \div 6=\frac{2}{3} \div \frac{6}{1}=\frac{2}{3} \times \frac{1}{6}\)