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Unit rates with fractions

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5502227
A transfer pump moves \(\frac{3}{4}\,\text{gallon}\) of water in \(\frac{1}{2}\,\text{minute}\). What is the unit rate in gallons per minute?

Hints

- A unit rate tells what happens for one unit of the second quantity. - Compare the amount of water with the fraction of a minute. - Check whether your answer should be greater or less than \(\frac{3}{4}\,\text{gallon}\).

Solution

1. Divide the amount of water by the time: \(\frac{3}{4}\div\frac{1}{2}=\frac{3}{2}\). 2. The pump moves \(\frac{3}{2}\,\text{gallons}\) in \(1\,\text{minute}\).

Answer

The unit rate is \(\frac{3}{2}\) gallons per minute, or \(1\frac{1}{2}\) gallons per minute.
5519257
A fabric cutter trims \(\frac{3}{5}\,\text{yard}\) of fabric in \(\frac{2}{3}\,\text{minute}\). Which quotient represents the unit rate in yards per minute? Do not simplify the quotient. a) \(\frac{3}{5}\div\frac{2}{3}\) b) \(\frac{2}{3}\div\frac{3}{5}\) c) \(\frac{3}{5}\cdot\frac{2}{3}\)

Hints

- Read the requested units in the order they are written. - Decide which given quantity should be divided by which other quantity. - You only need to choose the correct quotient; you do not need to calculate it.

Solution

1. A rate in yards per minute compares the number of yards with the number of minutes. 2. Therefore, divide the fabric length by the time: \(\frac{3}{5}\div\frac{2}{3}\). 3. Choice a) has the quantities in the required order.

Answer

a) \(\frac{3}{5}\div\frac{2}{3}\)
5502237
A kayaker travels \(2\frac{5}{8}\,\text{miles}\) in \(\frac{3}{4}\,\text{hour}\) at a constant rate. Find the kayaker's speed in miles per hour.

Hints

- Decide which quantity belongs in the numerator of the rate you are asked to find. - Rewrite the mixed number in a form that is convenient for division. - Your result should describe the distance for one full hour.

Solution

1. Rewrite \(2\frac{5}{8}=\frac{21}{8}\). 2. Divide distance by time: \(\frac{21}{8}\div\frac{3}{4}=\frac{21}{8}\cdot\frac{4}{3}=\frac{7}{2}\). 3. The speed is \(\frac{7}{2}\) miles per hour.

Answer

The kayaker's speed is \(\frac{7}{2}\) miles per hour, or \(3\frac{1}{2}\) miles per hour.
5502257
A paint roller covers \(\frac{5}{6}\,\text{square yard}\) using \(\frac{2}{3}\,\text{pint}\) of paint. At the same rate, how many square yards does the roller cover per pint?

Hints

- Keep track of which quantity should be “per pint.” - The given paint amount is less than one pint, so think about how the covered area should change when scaled to one pint. - Check that the units of your quotient match the requested unit rate.

Solution

1. Divide area by paint used: \(\frac{5}{6}\div\frac{2}{3}=\frac{5}{6}\cdot\frac{3}{2}=\frac{5}{4}\). 2. The roller covers \(\frac{5}{4}\) square yards per pint.

Answer

The unit rate is \(\frac{5}{4}\) square yards per pint, or \(1\frac{1}{4}\) square yards per pint.
5502287
The two number lines describe the same constant-speed trip. Number line a) shows time in hours, and number line b) shows distance in miles. The marked endpoints correspond to the same moment. Find the unit rate in miles per hour.
Figure for problem 550228

Hints

- Identify the two marked quantities that correspond to each other. - The requested rate asks for the distance associated with one full hour. - Use the labels on the number lines rather than estimating between ticks.

Solution

1. The diagram shows \(\frac{3}{4}\,\text{hour}\) corresponding to \(2\frac{1}{4}\,\text{miles}\). 2. Divide distance by time: \(\frac{9}{4}\div\frac{3}{4}=3\). 3. The unit rate is \(3\) miles per hour.

Answer

The unit rate is \(3\) miles per hour.
5502307
A full bar represents \(1\,\text{pound}\) of birdseed. The shaded part shows how much a dispenser releases in \(\frac{1}{4}\,\text{minute}\). Find the dispenser's unit rate in pounds per minute.
Figure for problem 550230

Hints

- First determine what fraction of the full bar is shaded. - Relate that shaded amount to the fraction of a minute in the problem. - The requested rate should describe one full minute.

Solution

1. The bar has \(8\) equal parts and \(5\) are shaded, so the dispenser releases \(\frac{5}{8}\,\text{pound}\) in \(\frac{1}{4}\,\text{minute}\). 2. Divide amount by time: \(\frac{5}{8}\div\frac{1}{4}=\frac{5}{2}\).

Answer

The unit rate is \(\frac{5}{2}\) pounds per minute, or \(2\frac{1}{2}\) pounds per minute.
5502217
A bread recipe is scaled proportionally. The table shows the amount of flour needed for several batch sizes. <table><tr><td>Batch size \(b\)</td><td>\(\frac{1}{4}\)</td><td>\(\frac{1}{2}\)</td><td>\(\frac{3}{4}\)</td></tr><tr><td>Flour \(f\) in cups</td><td>\(\frac{3}{8}\)</td><td>\(\frac{3}{4}\)</td><td>\(\frac{9}{8}\)</td></tr></table> a) Show that the relationship is proportional. b) Find the constant of proportionality in cups of flour per full batch. c) How much flour is needed for \(1\) full batch?

Hints

- Compare the flour amount with the corresponding fraction of a batch in every column. - Dividing fractions can reveal the amount needed for one full batch. - Use the unit-rate interpretation to answer the final part.

Solution

1. The ratios are \(\frac{3}{8}\div\frac{1}{4}=\frac{3}{2}\), \(\frac{3}{4}\div\frac{1}{2}=\frac{3}{2}\), and \(\frac{9}{8}\div\frac{3}{4}=\frac{3}{2}\). 2. Since the ratios are equal, the relationship is proportional with constant \(k=\frac{3}{2}\) cups per full batch. 3. For \(1\) full batch, the recipe needs \(\frac{3}{2}\,\text{cups}\) of flour.

Answer

a) The relationship is proportional because each ratio \(\frac{f}{b}\) equals \(\frac{3}{2}\). b) \(\frac{3}{2}\) cups per full batch. c) \(\frac{3}{2}\,\text{cups}\)
5502247
Two hikers move at constant speeds. Hiker A walks \(1\frac{7}{8}\,\text{miles}\) in \(\frac{1}{2}\,\text{hour}\). Hiker B walks \(2\frac{2}{3}\,\text{miles}\) in \(\frac{3}{4}\,\text{hour}\). Which hiker has the greater unit rate in miles per hour? Justify your comparison exactly, without rounding.

Hints

- Find a miles-per-hour rate for each hiker before comparing them. - Keep the rates as fractions so no information is lost to rounding. - Two fractions can be compared without converting them to decimals.

Solution

1. Hiker A's unit rate is \(\frac{15}{8}\div\frac{1}{2}=\frac{15}{4}\) miles per hour. 2. Hiker B's unit rate is \(\frac{8}{3}\div\frac{3}{4}=\frac{32}{9}\) miles per hour. 3. Compare \(\frac{15}{4}\) and \(\frac{32}{9}\): \(15\cdot9=135\) and \(32\cdot4=128\), so \(\frac{15}{4}>\frac{32}{9}\).

Answer

Hiker A has the greater unit rate: \(\frac{15}{4}\) miles per hour compared with \(\frac{32}{9}\) miles per hour for Hiker B.
5502267
A cutting machine trims \(\frac{7}{12}\,\text{meter}\) of material in \(\frac{1}{6}\,\text{minute}\). A student says the unit rate is \(\frac{2}{7}\) meter per minute because \(\frac{1}{6}\div\frac{7}{12}=\frac{2}{7}\). Explain the error and find the correct unit rate in meters per minute.

Hints

- Read the requested units in order before choosing a quotient. - Ask which quantity should be measured for exactly one minute. - Compare the student's quotient units with the units named in the question.

Solution

1. The requested rate is meters per minute, so material length must be divided by time, not time by length. 2. Compute \(\frac{7}{12}\div\frac{1}{6}=\frac{7}{12}\cdot6=\frac{7}{2}\). 3. The student found the reciprocal rate, minutes per meter, rather than meters per minute.

Answer

The correct unit rate is \(\frac{7}{2}\) meters per minute, or \(3\frac{1}{2}\) meters per minute. The student reversed the two quantities.
5502277
A drink recipe uses \(\frac{5}{8}\,\text{cup}\) of concentrate for \(\frac{3}{4}\) of a full batch. a) Find the unit rate in cups of concentrate per full batch. b) Find the reciprocal unit rate in batches per cup of concentrate. c) Explain how the two unit rates are related.

Hints

- The word “per” tells you which quantity should correspond to one unit. - Parts a) and b) use the same two quantities in opposite orders. - Compare the two quotients after you simplify them.

Solution

1. Cups per batch: \(\frac{5}{8}\div\frac{3}{4}=\frac{5}{6}\). 2. Batches per cup: \(\frac{3}{4}\div\frac{5}{8}=\frac{6}{5}\). 3. The two unit rates are reciprocals because the order of the two quantities is reversed.

Answer

a) \(\frac{5}{6}\) cup per batch b) \(\frac{6}{5}\) batches per cup c) The two rates are reciprocals.
5502297
A sculpting class uses \(\frac{3}{4}\,\text{pound}\) of clay to make \(\frac{2}{5}\) of a full-size model. The amount of clay is proportional to the number of full-size models. a) Find the pounds of clay needed per full-size model. b) How many pounds of clay are needed for \(3\) full-size models?

Hints

- First determine the amount associated with one complete model. - The given amount of clay corresponds to only part of a model. - After finding the unit rate, scale it to the requested number of models.

Solution

1. Find the unit rate: \(\frac{3}{4}\div\frac{2}{5}=\frac{3}{4}\cdot\frac{5}{2}=\frac{15}{8}\) pounds per model. 2. For \(3\) models, \(3\cdot\frac{15}{8}=\frac{45}{8}\) pounds.

Answer

a) \(\frac{15}{8}\) pounds per model, or \(1\frac{7}{8}\) pounds per model b) \(\frac{45}{8}\) pounds, or \(5\frac{5}{8}\) pounds
5502317
A test strip of flooring covers \(\frac{15}{16}\,\text{square yard}\) using \(\frac{3}{4}\,\text{pint}\) of finish. The coverage is proportional to the amount of finish. a) Find the unit rate in square yards per pint. b) How many pints are needed to cover \(2\frac{1}{2}\,\text{square yards}\)?

Hints

- Start by finding the coverage for one pint. - In part b, decide whether the known area should be multiplied by or divided by the coverage rate. - Keep the area units consistent throughout.

Solution

1. Unit rate: \(\frac{15}{16}\div\frac{3}{4}=\frac{5}{4}\) square yards per pint. 2. For \(2\frac{1}{2}=\frac{5}{2}\) square yards, divide by the unit rate: \(\frac{5}{2}\div\frac{5}{4}=2\) pints.

Answer

a) \(\frac{5}{4}\) square yards per pint b) \(2\,\text{pints}\)
5502327
An ice maker produces \(\frac{11}{16}\,\text{pound}\) of ice in \(\frac{3}{8}\,\text{hour}\) at a constant rate. a) Find the unit rate in pounds per hour. b) At that rate, how many pounds of ice will it produce in \(1\frac{1}{2}\,\text{hours}\)?

Hints

- Convert the given production information into an amount for one hour. - Keep the rate exact rather than rounding it. - Use the rate again for the longer time in part b.

Solution

1. Unit rate: \(\frac{11}{16}\div\frac{3}{8}=\frac{11}{6}\) pounds per hour. 2. For \(1\frac{1}{2}=\frac{3}{2}\) hours, \(\frac{11}{6}\cdot\frac{3}{2}=\frac{11}{4}\) pounds.

Answer

a) \(\frac{11}{6}\) pounds per hour b) \(\frac{11}{4}\) pounds, or \(2\frac{3}{4}\) pounds

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