Solved on Feb 21, 2024

Find products of fractions and whole numbers. Determine if fraction-based statements are true or false. Calculate miles ridden and explain equivalence of fraction multiplication and division.
1a) $\frac{1}{4}$ of $6=1.5$ 1b) $\frac{1}{5} \times 30=6$ 1c) $\frac{1}{3}$ of $27=9$ 1d) $\frac{3}{4}$ of $6=4.5$ 1e) $\frac{4}{5} \times 30=24$ 1f) $\frac{2}{3} \times 27=18$
2a) $\frac{1}{4} \times 9=2.25$ (False) 2b) $\frac{3}{5}$ of $25=15$ (True) 2c) $\frac{2}{5}$ of $15=6$ (False) 2d) $18 \times \frac{1}{5}=3.6$ (False) 2e) $\frac{2}{6} \times 24=8$ (True) 2f) $17 \times \frac{1}{3}=\frac{17}{3}$ (True)
3) Pete rode $\frac{2}{3}$ of 150 miles, which is 100 miles.
4) Kim is correct. $\frac{1}{4} \times 12=3$ , which is the same as dividing 12 by 4.

STEP 1

Assumptions
1. To find a fraction of a whole number, we multiply the fraction by the number.
2. To verify if a statement is true or false, we calculate the left side of the equation and compare it with the right side.
3. Multiplying by a fraction is equivalent to dividing by its reciprocal.

STEP 2

Find $\frac{1}{4}$ of $6$ by multiplying the fraction by the number.
$\frac{1}{4} \times 6$

STEP 3

Calculate the product for part a.
$\frac{1}{4} \times 6 = \frac{6}{4} = \frac{3}{2} = 1.5$

STEP 4

Find $\frac{1}{5} \times 30$ by multiplying the fraction by the number.
$\frac{1}{5} \times 30$

STEP 5

Calculate the product for part b.
$\frac{1}{5} \times 30 = \frac{30}{5} = 6$

STEP 6

Find $\frac{1}{3}$ of $27$ by multiplying the fraction by the number.
$\frac{1}{3} \times 27$

STEP 7

Calculate the product for part c.
$\frac{1}{3} \times 27 = \frac{27}{3} = 9$

STEP 8

Find $\frac{3}{4}$ of $6$ by multiplying the fraction by the number.
$\frac{3}{4} \times 6$

STEP 9

Calculate the product for part d.
$\frac{3}{4} \times 6 = \frac{18}{4} = \frac{9}{2} = 4.5$

STEP 10

Find $\frac{4}{5} \times 30$ by multiplying the fraction by the number.
$\frac{4}{5} \times 30$

STEP 11

Calculate the product for part e.
$\frac{4}{5} \times 30 = \frac{120}{5} = 24$

STEP 12

Find $\frac{2}{3} \times 27$ by multiplying the fraction by the number.
$\frac{2}{3} \times 27$

STEP 13

Calculate the product for part f.
$\frac{2}{3} \times 27 = \frac{54}{3} = 18$

STEP 14

Determine if $\frac{1}{4} \times 9=2 \frac{1}{4}$ is true or false.
$\frac{1}{4} \times 9$

STEP 15

Calculate the product for part 2a.
$\frac{1}{4} \times 9 = \frac{9}{4} = 2 \frac{1}{4}$
The statement is True.

STEP 16

Determine if $\frac{3}{5}$ of $25=15$ is true or false.
$\frac{3}{5} \times 25$

STEP 17

Calculate the product for part 2b.
$\frac{3}{5} \times 25 = \frac{75}{5} = 15$
The statement is True.

STEP 18

Determine if $\frac{2}{5}$ of $15=5 \frac{2}{5}$ is true or false.
$\frac{2}{5} \times 15$

STEP 19

Calculate the product for part 2c.
$\frac{2}{5} \times 15 = \frac{30}{5} = 6$
The statement is False.

STEP 20

Determine if $18 \times \frac{1}{5}=\frac{5}{18}$ is true or false.
$18 \times \frac{1}{5}$

STEP 21

Calculate the product for part 2d.
$18 \times \frac{1}{5} = \frac{18}{5} = 3 \frac{3}{5}$
The statement is False.

STEP 22

Determine if $\frac{2}{6} \times 24=14$ is true or false.
$\frac{2}{6} \times 24$

STEP 23

Calculate the product for part 2e.
$\frac{2}{6} \times 24 = \frac{48}{6} = 8$
The statement is False.

STEP 24

Determine if $17 \times \frac{1}{3}=\frac{17}{3}$ is true or false.
$17 \times \frac{1}{3}$

STEP 25

Calculate the product for part 2f.
$17 \times \frac{1}{3} = \frac{17}{3}$
The statement is True.

STEP 26

Calculate how many miles Pete rode on his dirt bike, which is $\frac{2}{3}$ of the 150-mile course.
$\frac{2}{3} \times 150$

STEP 27

Calculate the distance Pete rode.
$\frac{2}{3} \times 150 = \frac{300}{3} = 100$
Pete rode 100 miles.

STEP 28

Determine if multiplying $\frac{1}{4} \times 12$ is the same as dividing $12$ by $4$ .
$\frac{1}{4} \times 12$

STEP 29

Calculate the product.
$\frac{1}{4} \times 12 = \frac{12}{4} = 3$

STEP 30

Compare the result with dividing $12$ by $4$ .
$12 \div 4 = 3$
Since both operations yield the same result, we agree with Kim.
The solution to the problems are as follows: 1a. $1.5$ 1b. $6$ 1c. $9$ 1d. $4.5$ 1e. $24$ 1f. $18$ 2a. True 2b. True 2c. False 2d. False 2e. False 2f. True
3. Pete rode $100$ miles.
4. Agree with Kim; both operations yield the same result.

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STEP 1

Assumptions1. To find a fraction of a whole number, we multiply the fraction by the number.2. To verify if a statement is true or false, we calculate the left side of the equation and compare it with the right side.3. Multiplying by a fraction is equivalent to dividing by its reciprocal.

STEP 2

Find 14\frac{1}{4}41​ of 666 by multiplying the fraction by the number.14×6\frac{1}{4} \times 641​×6

STEP 3

Calculate the product for part a.14×6=64=32=1.5\frac{1}{4} \times 6 = \frac{6}{4} = \frac{3}{2} = 1.541​×6=46​=23​=1.5

STEP 4

Find 15×30\frac{1}{5} \times 3051​×30 by multiplying the fraction by the number.15×30\frac{1}{5} \times 3051​×30

STEP 5

Calculate the product for part b.15×30=305=6\frac{1}{5} \times 30 = \frac{30}{5} = 651​×30=530​=6

STEP 6

Find 13\frac{1}{3}31​ of 272727 by multiplying the fraction by the number.13×27\frac{1}{3} \times 2731​×27

STEP 7

Calculate the product for part c.13×27=273=9\frac{1}{3} \times 27 = \frac{27}{3} = 931​×27=327​=9

STEP 8

Find 34\frac{3}{4}43​ of 666 by multiplying the fraction by the number.34×6\frac{3}{4} \times 643​×6

STEP 9

Calculate the product for part d.34×6=184=92=4.5\frac{3}{4} \times 6 = \frac{18}{4} = \frac{9}{2} = 4.543​×6=418​=29​=4.5

STEP 10

Find 45×30\frac{4}{5} \times 3054​×30 by multiplying the fraction by the number.45×30\frac{4}{5} \times 3054​×30

STEP 11

Calculate the product for part e.45×30=1205=24\frac{4}{5} \times 30 = \frac{120}{5} = 2454​×30=5120​=24

STEP 12

Find 23×27\frac{2}{3} \times 2732​×27 by multiplying the fraction by the number.23×27\frac{2}{3} \times 2732​×27

STEP 13

Calculate the product for part f.23×27=543=18\frac{2}{3} \times 27 = \frac{54}{3} = 1832​×27=354​=18

STEP 14

Determine if 14×9=214\frac{1}{4} \times 9=2 \frac{1}{4}41​×9=241​ is true or false.14×9\frac{1}{4} \times 941​×9

STEP 15

Calculate the product for part 2a.14×9=94=214\frac{1}{4} \times 9 = \frac{9}{4} = 2 \frac{1}{4}41​×9=49​=241​The statement is True.

STEP 16

Determine if 35\frac{3}{5}53​ of 25=1525=1525=15 is true or false.35×25\frac{3}{5} \times 2553​×25

STEP 17

Calculate the product for part 2b.35×25=755=15\frac{3}{5} \times 25 = \frac{75}{5} = 1553​×25=575​=15The statement is True.

STEP 18

Determine if 25\frac{2}{5}52​ of 15=52515=5 \frac{2}{5}15=552​ is true or false.25×15\frac{2}{5} \times 1552​×15

STEP 19

Calculate the product for part 2c.25×15=305=6\frac{2}{5} \times 15 = \frac{30}{5} = 652​×15=530​=6The statement is False.

STEP 20

Determine if 18×15=51818 \times \frac{1}{5}=\frac{5}{18}18×51​=185​ is true or false.18×1518 \times \frac{1}{5}18×51​

STEP 21

Calculate the product for part 2d.18×15=185=33518 \times \frac{1}{5} = \frac{18}{5} = 3 \frac{3}{5}18×51​=518​=353​The statement is False.

STEP 22

Determine if 26×24=14\frac{2}{6} \times 24=1462​×24=14 is true or false.26×24\frac{2}{6} \times 2462​×24

STEP 23

Calculate the product for part 2e.26×24=486=8\frac{2}{6} \times 24 = \frac{48}{6} = 862​×24=648​=8The statement is False.

STEP 24

Determine if 17×13=17317 \times \frac{1}{3}=\frac{17}{3}17×31​=317​ is true or false.17×1317 \times \frac{1}{3}17×31​

STEP 25

Calculate the product for part 2f.17×13=17317 \times \frac{1}{3} = \frac{17}{3}17×31​=317​The statement is True.

STEP 26

Calculate how many miles Pete rode on his dirt bike, which is 23\frac{2}{3}32​ of the 150-mile course.23×150\frac{2}{3} \times 15032​×150

STEP 27

Calculate the distance Pete rode.23×150=3003=100\frac{2}{3} \times 150 = \frac{300}{3} = 10032​×150=3300​=100Pete rode 100 miles.

STEP 28

Determine if multiplying 14×12\frac{1}{4} \times 1241​×12 is the same as dividing 121212 by 444.14×12\frac{1}{4} \times 1241​×12

STEP 29

Calculate the product.14×12=124=3\frac{1}{4} \times 12 = \frac{12}{4} = 341​×12=412​=3

STEP 30

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Assumptions
1. To find a fraction of a whole number, we multiply the fraction by the number.
2. To verify if a statement is true or false, we calculate the left side of the equation and compare it with the right side.
3. Multiplying by a fraction is equivalent to dividing by its reciprocal.

Find $\frac{1}{4}$ of $6$ by multiplying the fraction by the number.
$\frac{1}{4} \times 6$

Calculate the product for part a.
$\frac{1}{4} \times 6 = \frac{6}{4} = \frac{3}{2} = 1.5$

Find $\frac{1}{5} \times 30$ by multiplying the fraction by the number.
$\frac{1}{5} \times 30$

Calculate the product for part b.
$\frac{1}{5} \times 30 = \frac{30}{5} = 6$

Find $\frac{1}{3}$ of $27$ by multiplying the fraction by the number.
$\frac{1}{3} \times 27$

Calculate the product for part c.
$\frac{1}{3} \times 27 = \frac{27}{3} = 9$

Find $\frac{3}{4}$ of $6$ by multiplying the fraction by the number.
$\frac{3}{4} \times 6$

Calculate the product for part d.
$\frac{3}{4} \times 6 = \frac{18}{4} = \frac{9}{2} = 4.5$

Find $\frac{4}{5} \times 30$ by multiplying the fraction by the number.
$\frac{4}{5} \times 30$

Calculate the product for part e.
$\frac{4}{5} \times 30 = \frac{120}{5} = 24$

Find $\frac{2}{3} \times 27$ by multiplying the fraction by the number.
$\frac{2}{3} \times 27$

Calculate the product for part f.
$\frac{2}{3} \times 27 = \frac{54}{3} = 18$

Determine if $\frac{1}{4} \times 9=2 \frac{1}{4}$ is true or false.
$\frac{1}{4} \times 9$

Calculate the product for part 2a.
$\frac{1}{4} \times 9 = \frac{9}{4} = 2 \frac{1}{4}$
The statement is True.

Determine if $\frac{3}{5}$ of $25=15$ is true or false.
$\frac{3}{5} \times 25$

Calculate the product for part 2b.
$\frac{3}{5} \times 25 = \frac{75}{5} = 15$
The statement is True.

Determine if $\frac{2}{5}$ of $15=5 \frac{2}{5}$ is true or false.
$\frac{2}{5} \times 15$

Calculate the product for part 2c.
$\frac{2}{5} \times 15 = \frac{30}{5} = 6$
The statement is False.

Determine if $18 \times \frac{1}{5}=\frac{5}{18}$ is true or false.
$18 \times \frac{1}{5}$

Calculate the product for part 2d.
$18 \times \frac{1}{5} = \frac{18}{5} = 3 \frac{3}{5}$
The statement is False.

Determine if $\frac{2}{6} \times 24=14$ is true or false.
$\frac{2}{6} \times 24$

Calculate the product for part 2e.
$\frac{2}{6} \times 24 = \frac{48}{6} = 8$
The statement is False.

Determine if $17 \times \frac{1}{3}=\frac{17}{3}$ is true or false.
$17 \times \frac{1}{3}$

Calculate the product for part 2f.
$17 \times \frac{1}{3} = \frac{17}{3}$
The statement is True.

Calculate how many miles Pete rode on his dirt bike, which is $\frac{2}{3}$ of the 150-mile course.
$\frac{2}{3} \times 150$

Calculate the distance Pete rode.
$\frac{2}{3} \times 150 = \frac{300}{3} = 100$
Pete rode 100 miles.

Determine if multiplying $\frac{1}{4} \times 12$ is the same as dividing $12$ by $4$ .
$\frac{1}{4} \times 12$

Calculate the product.
$\frac{1}{4} \times 12 = \frac{12}{4} = 3$