1
A printing company uses 8 sheets of cardstock to make 1 greeting card package. If the company used 144 sheets of cardstock, how many greeting card packages did it make?
Ratios, rates, proportional relationships, and units: worksheet with answers
Work through the questions first, then open the answer key and explanations to check your work.
2
A streaming service downloads videos at a constant rate of 18 megabytes per second. How many megabytes are downloaded in 7 seconds?
3
A bakery packs 8 muffins in each box. If the bakery has 56 muffins to pack, how many boxes are needed?
4
A bakery uses a machine that fills 18 boxes with cookies every 3 minutes at a constant rate. At this rate, how many boxes will the machine fill in 11 minutes?
5
If \(\frac{x}{y}=4\) and \(\frac{9x}{ky}=4\), what is the value of \(k\)?
6
A sports drink is mixed so that the ratio of cups of concentrate to cups of water is \(3:8\). If the amount of water used is increased by 16 cups, by how many cups must the amount of concentrate increase to keep the mixture in the same ratio?
7
A drone transfers data at a rate of 3.5 gigabytes per minute. If 1 gigabyte equals 1,000 megabytes, how many megabytes per minute does the drone transfer?
8
A packaging machine fills \(37.5\) cartons per minute. Each carton contains 24 juice pouches. At what rate, in juice pouches per minute, does the machine fill juice pouches?
9
A solid glass paperweight is shaped like a cylinder with a radius of \(2.5\) centimeters and a height of \(8\) centimeters. The glass has a density of \(2.4\) grams per cubic centimeter. What is the mass of the paperweight, in grams, rounded to the nearest whole number?
10
A conveyor belt moves packages at a constant rate. Conveyor A moves \(9p\) packages in \(t\) minutes. Conveyor A operates at 3 times the rate of Conveyor B.
At Conveyor B's constant rate, how many minutes would it take to move \(27p\) packages?
11
A cube made from a ceramic material has a mass of 1,420 grams. The material has a density of 58 grams per cubic centimeter. What is the approximate edge length of the cube, in centimeters, rounded to the nearest hundredth?
12
If \(8\left(\frac{m}{n}\right)=5.6\) and \(\frac{m}{nt}=0.35\), what is the value of \(t\)?
Answer key and explanations
1. Answer
18
Each greeting card package requires 8 sheets of cardstock. To find the number of packages made from 144 sheets, divide the total number of sheets by the number of sheets per package:
\(144 \div 8 = 18\)
The company made 18 greeting card packages.
2. Answer
126
Multiply the download rate by the number of seconds: \(18 \times 7 = 126\). Therefore, 126 megabytes are downloaded in 7 seconds.
3. Answer
7
Each box holds 8 muffins. To find the number of boxes needed for 56 muffins, divide 56 by 8.
\(56 \div 8 = 7\)
So, 7 boxes are needed.
4. Answer
66
The machine fills \(18 \div 3 = 6\) boxes per minute.
In 11 minutes, the machine will fill \(6 \times 11 = 66\) boxes.
Therefore, the correct answer is \(66\).
5. Answer
9
Since \(\frac{x}{y}=4\), substitute that ratio into the second equation:
\(\frac{9x}{ky}=\frac{9}{k}\cdot\frac{x}{y}\)
This becomes \(\frac{9}{k}\cdot 4=4\).
Divide both sides by 4:
\(\frac{9}{k}=1\)
So \(k=9\).
6. Answer
Increase by 6 cups
The ratio of concentrate to water is \(3:8\), which means the amount of concentrate must change by \(\frac{3}{8}\) of any change in the amount of water. Since the water increases by 16 cups, the concentrate must increase by \(\frac{3}{8}\times 16=6\) cups. Therefore, the correct answer is that the concentrate must increase by 6 cups.
7. Answer
3500
To convert gigabytes per minute to megabytes per minute, multiply by 1,000 because each gigabyte contains 1,000 megabytes.
\(3.5 \times 1000 = 3500\)
The drone transfers 3,500 megabytes per minute.
8. Answer
900
Multiply the number of cartons filled each minute by the number of juice pouches in each carton:
\(37.5 \times 24 = 900\)
The cartons unit cancels, leaving juice pouches per minute:
\((37.5\ \text{cartons/minute})\times(24\ \text{pouches/carton})=900\ \text{pouches/minute}\)
So the correct response is \(900\).
9. Answer
377
First find the volume of the cylinder using \(V=\pi r^{2}h\).
\(V=\pi(2.5)^{2}(8)=\pi(6.25)(8)=50\pi\) cubic centimeters.
Now multiply the volume by the density:
\((50\pi)(2.4)=120\pi\approx 376.99\)
Rounded to the nearest whole number, the mass is \(377\) grams.
10. Answer
\(9t\)
Conveyor A moves \(9p\) packages in \(t\) minutes, so its rate is \(\frac{9p}{t}\) packages per minute.
Since Conveyor A operates at 3 times the rate of Conveyor B, Conveyor B's rate is \(\frac{1}{3}\cdot\frac{9p}{t}=\frac{3p}{t}\) packages per minute.
The time needed for Conveyor B to move \(27p\) packages is
\(\frac{27p}{\frac{3p}{t}}=27p\cdot\frac{t}{3p}=9t\).
Therefore, the correct answer is \(9t\).
11. Answer
2.90 cm
Density is defined as mass divided by volume, so the volume of the cube is
\(\frac{1420}{58}\approx 24.48\) cubic centimeters.
For a cube with edge length \(s\), the volume formula is \(s^3\). Therefore,
\(s^3\approx 24.48\)
Taking the cube root gives
\(s\approx \sqrt[3]{24.48}\approx 2.90\).
So the edge length of the cube is approximately 2.90 centimeters.
12. Answer
2 / 2.0
From \(8\left(\frac{m}{n}\right)=5.6\), divide both sides by 8:
\(\frac{m}{n}=\frac{5.6}{8}=0.7\)
Substitute this value into the second equation:
\(\frac{m}{nt}=0.35\) becomes \(\frac{0.7}{t}=0.35\).
Multiply both sides by \(t\):
\(0.7=0.35t\)
Then divide by \(0.35\):
\(t=2\).