Question

apply and use rates, proportions and percents.

1. circle each proportion that can be used to solve each problem correctly for a-c. for (d-f) fill in the blanks.

a. ben can type 296 words in 8 minutes. at this rate, how many words can he type in 20 minutes?
$\frac{8}{296} = \frac{20}{w}$ $\frac{8}{20} = \frac{296}{w}$
$\frac{296}{w} = \frac{20}{8}$ $\frac{20}{8} = \frac{w}{296}$

b. martin can travel 174 miles in 3 hours. at this rate, how far can he travel in 7 hours?
$\frac{174}{3} = \frac{m}{7}$ $\frac{174}{m} = \frac{3}{7}$
$\frac{3}{174} = \frac{7}{m}$ $\frac{3}{7} = \frac{174}{m}$

c. jamie can text 312 words in 4 minutes. at this rate, how many words can jamie text in 15 minutes?
$\frac{15}{4} = \frac{312}{t}$ $\frac{4}{312} = \frac{15}{t}$
$\frac{312}{4} = \frac{15}{t}$ $\frac{4}{15} = \frac{t}{312}$

d. morgan typed 2 pages in 15 minutes, at this rate how long will it morgan to type 10 pages?
$\frac{\square}{\square} = \frac{\square}{\square}$

e. terry makes 5 out of every 8 free throws that he shoots. if terry shoots 32 free throws how many will he make?
$\frac{5}{8} = \frac{x}{32}$ (with multiplication arrows)

f. if 4 peaches weighs 54 ounces, then how many peaches weigh 27 ounces?
$\frac{4}{54} = \frac{x}{27}$ (with division arrows)

Explanation:

Part A

Step1: Understand the rate

Ben's typing rate is constant, so words per minute should be equal. So \(\frac{\text{words}}{\text{minutes}}\) should be equal. So \(\frac{296}{8}=\frac{w}{20}\) or rearranged as \(\frac{296}{w}=\frac{20}{8}\) (cross - multiplication equivalent) and \(\frac{8}{296}=\frac{20}{w}\) (time over words ratio). The proportion \(\frac{8}{296}=\frac{20}{w}\) is correct (time1/words1 = time2/words2), \(\frac{296}{w}=\frac{20}{8}\) is also correct (words1/time1=words2/time2 rearranged), and \(\frac{20}{8}=\frac{w}{296}\) is wrong (20/8 is time ratio, w/296 is words ratio which is not equal). \(\frac{8}{20}=\frac{296}{w}\) is wrong (time ratio 8/20 and words ratio 296/w not equal). So correct proportions are \(\frac{8}{296}=\frac{20}{w}\) and \(\frac{296}{w}=\frac{20}{8}\)

Part B

Step1: Understand the rate

Martin's speed is constant, speed=\(\frac{\text{distance}}{\text{time}}\). So \(\frac{174}{3}=\frac{m}{7}\) (distance1/time1 = distance2/time2) and \(\frac{3}{174}=\frac{7}{m}\) (time1/distance1 = time2/distance2, cross - multiplication gives same as first). The proportion \(\frac{174}{m}=\frac{3}{7}\) is wrong (distance1/distance2 = time1/time2 which is not correct) and \(\frac{3}{7}=\frac{174}{m}\) is wrong. So correct proportions are \(\frac{174}{3}=\frac{m}{7}\) and \(\frac{3}{174}=\frac{7}{m}\)

Part C

Step1: Understand the rate

Jamie's typing rate is constant, \(\frac{\text{words}}{\text{minutes}}\) is constant. So \(\frac{312}{4}=\frac{t}{15}\) (words1/time1 = words2/time2) and \(\frac{4}{312}=\frac{15}{t}\) is wrong (time1/words1 = time2/words2? No, 4/312 is time per word, 15/t is time per word. Wait, \(\frac{4}{312}=\frac{15}{t}\) cross - multiplies to \(4t = 312\times15\), and \(\frac{312}{4}=\frac{t}{15}\) cross - multiplies to \(312\times15=4t\). Wait, actually \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\) are equivalent? Wait no, \(\frac{312}{4}\) is words per minute, \(\frac{t}{15}\) is words per minute. \(\frac{15}{4}\) is time ratio, \(\frac{312}{t}\) is words ratio. So \(\frac{15}{4}=\frac{312}{t}\) is wrong. Wait, let's re - express: Let \(t\) be the number of words in 15 minutes. Rate is \(\frac{312}{4}=\frac{t}{15}\), so correct proportion is \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\) is wrong (because \(\frac{4}{312}\) is minutes per word, \(\frac{15}{t}\) is minutes per word, so actually \(\frac{4}{312}=\frac{15}{t}\) is correct. Wait, cross - multiply \(\frac{312}{4}=\frac{t}{15}\) gives \(312\times15 = 4t\), \(\frac{4}{312}=\frac{15}{t}\) gives \(4t=312\times15\). So both \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\) are correct? Wait no, the first option \(\frac{15}{4}=\frac{312}{t}\) cross - multiplies to \(15t = 4\times312\), which is different. So correct proportions are \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\)

Part D

Step1: Set up the proportion

The rate is pages per minute. So \(\frac{\text{pages}}{\text{minutes}}\) is constant. Morgan types 2 pages in 15 minutes, and \(x\) minutes for 10 pages. So \(\frac{2}{15}=\frac{10}{x}\) (or \(\frac{15}{2}=\frac{x}{10}\) but the blank is \(\frac{\square}{\square}=\frac{\square}{\square}\), so \(\frac{2}{15}=\frac{10}{x}\) (filling the blanks: \(\frac{2}{15}=\frac{10}{x}\), so numerator and denominator: \(\frac{2}{15}=\frac{10}{75}\) (solving: \(2x = 15\times10\), \(x = 75\))

Part E

Step1: Solve the proportion

We have \(\frac{5}{8}=\frac{x}{32}\). Cross - multiply: \(8x=5\times32\), \(8x = 160\), \(x=\frac{160}{8}=20\). So we mu…

Answer:

Step1: Understand the rate

Ben's typing rate is constant, so words per minute should be equal. So \(\frac{\text{words}}{\text{minutes}}\) should be equal. So \(\frac{296}{8}=\frac{w}{20}\) or rearranged as \(\frac{296}{w}=\frac{20}{8}\) (cross - multiplication equivalent) and \(\frac{8}{296}=\frac{20}{w}\) (time over words ratio). The proportion \(\frac{8}{296}=\frac{20}{w}\) is correct (time1/words1 = time2/words2), \(\frac{296}{w}=\frac{20}{8}\) is also correct (words1/time1=words2/time2 rearranged), and \(\frac{20}{8}=\frac{w}{296}\) is wrong (20/8 is time ratio, w/296 is words ratio which is not equal). \(\frac{8}{20}=\frac{296}{w}\) is wrong (time ratio 8/20 and words ratio 296/w not equal). So correct proportions are \(\frac{8}{296}=\frac{20}{w}\) and \(\frac{296}{w}=\frac{20}{8}\)

Part B

Step1: Understand the rate

Martin's speed is constant, speed=\(\frac{\text{distance}}{\text{time}}\). So \(\frac{174}{3}=\frac{m}{7}\) (distance1/time1 = distance2/time2) and \(\frac{3}{174}=\frac{7}{m}\) (time1/distance1 = time2/distance2, cross - multiplication gives same as first). The proportion \(\frac{174}{m}=\frac{3}{7}\) is wrong (distance1/distance2 = time1/time2 which is not correct) and \(\frac{3}{7}=\frac{174}{m}\) is wrong. So correct proportions are \(\frac{174}{3}=\frac{m}{7}\) and \(\frac{3}{174}=\frac{7}{m}\)

Part C

Step1: Understand the rate

Jamie's typing rate is constant, \(\frac{\text{words}}{\text{minutes}}\) is constant. So \(\frac{312}{4}=\frac{t}{15}\) (words1/time1 = words2/time2) and \(\frac{4}{312}=\frac{15}{t}\) is wrong (time1/words1 = time2/words2? No, 4/312 is time per word, 15/t is time per word. Wait, \(\frac{4}{312}=\frac{15}{t}\) cross - multiplies to \(4t = 312\times15\), and \(\frac{312}{4}=\frac{t}{15}\) cross - multiplies to \(312\times15=4t\). Wait, actually \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\) are equivalent? Wait no, \(\frac{312}{4}\) is words per minute, \(\frac{t}{15}\) is words per minute. \(\frac{15}{4}\) is time ratio, \(\frac{312}{t}\) is words ratio. So \(\frac{15}{4}=\frac{312}{t}\) is wrong. Wait, let's re - express: Let \(t\) be the number of words in 15 minutes. Rate is \(\frac{312}{4}=\frac{t}{15}\), so correct proportion is \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\) is wrong (because \(\frac{4}{312}\) is minutes per word, \(\frac{15}{t}\) is minutes per word, so actually \(\frac{4}{312}=\frac{15}{t}\) is correct. Wait, cross - multiply \(\frac{312}{4}=\frac{t}{15}\) gives \(312\times15 = 4t\), \(\frac{4}{312}=\frac{15}{t}\) gives \(4t=312\times15\). So both \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\) are correct? Wait no, the first option \(\frac{15}{4}=\frac{312}{t}\) cross - multiplies to \(15t = 4\times312\), which is different. So correct proportions are \(\frac{312}{4}=\frac{t}{15}\) and \(\frac{4}{312}=\frac{15}{t}\)

Part D

Step1: Set up the proportion

The rate is pages per minute. So \(\frac{\text{pages}}{\text{minutes}}\) is constant. Morgan types 2 pages in 15 minutes, and \(x\) minutes for 10 pages. So \(\frac{2}{15}=\frac{10}{x}\) (or \(\frac{15}{2}=\frac{x}{10}\) but the blank is \(\frac{\square}{\square}=\frac{\square}{\square}\), so \(\frac{2}{15}=\frac{10}{x}\) (filling the blanks: \(\frac{2}{15}=\frac{10}{x}\), so numerator and denominator: \(\frac{2}{15}=\frac{10}{75}\) (solving: \(2x = 15\times10\), \(x = 75\))

Part E

Step1: Solve the proportion

We have \(\frac{5}{8}=\frac{x}{32}\). Cross - multiply: \(8x=5\times32\), \(8x = 160\), \(x=\frac{160}{8}=20\). So we multiply 5 by \(32\div8 = 4\), so \(5\times4 = 20\), and multiply 8 by 4 (or divide 32 by 8). So the blanks are multiply by 4 (both top and bottom, since \(\frac{5\times4}{8\times4}=\frac{20}{32}\))

Part F

Step1: Solve the proportion

We have \(\frac{4}{54}=\frac{x}{27}\). Cross - multiply: \(54x = 4\times27\), \(54x=108\), \(x = \frac{108}{54}=2\). We can see that 54 is divided by 2 to get 27, so we divide 4 by 2 to get \(x\). So the operation is divide by 2 (both numerator and denominator, \(\frac{4\div2}{54\div2}=\frac{2}{27}\))

Final Answers (for each part):

• Part A: Circle \(\boldsymbol{\frac{8}{296}=\frac{20}{w}}\) and \(\boldsymbol{\frac{296}{w}=\frac{20}{8}}\)
• Part B: Circle \(\boldsymbol{\frac{174}{3}=\frac{m}{7}}\) and \(\boldsymbol{\frac{3}{174}=\frac{7}{m}}\)
• Part C: Circle \(\boldsymbol{\frac{312}{4}=\frac{t}{15}}\) and \(\boldsymbol{\frac{4}{312}=\frac{15}{t}}\) (Note: Wait, earlier miscalculation, \(\frac{312}{4}\) is words per minute, \(\frac{t}{15}\) is words per minute, so \(\frac{312}{4}=\frac{t}{15}\) is correct. \(\frac{4}{312}\) is minutes per word, \(\frac{15}{t}\) is minutes per word, so \(\frac{4}{312}=\frac{15}{t}\) is also correct. The other two are wrong)
• Part D: \(\boldsymbol{\frac{2}{15}=\frac{10}{75}}\) (or \(\frac{2}{15}=\frac{10}{x}\) with \(x = 75\))
• Part E: \(x=\boldsymbol{20}\) (multiply by 4)
• Part F: \(x=\boldsymbol{2}\) (divide by 2)