Grade 7, Unit 2 - Practice Problems (2024)

Which one of these shapes is not like the others? Explain what makes it different by representing each width and height pair with a ratio.

In one version of a trail mix, there are 3 cups of peanuts mixed with 2 cups of raisins. In another version of trail mix, there are 4.5 cups of peanuts mixed with 3 cups of raisins. Are the ratios equivalent for the two mixes? Explain your reasoning.

For each object, choose an appropriate scale for a drawing that fits on a regular sheet of paper. Not all of the scales on the list will be used.

Which scale is equivalent to 1 cm to 1 km?

1. 1 to 1000
2. 10,000 to 1
3. 1 to 100,000
4. 100,000 to 1
5. 1 to 1,000,000

1. Find 3 different ratios that are equivalent to $7:3$.
2. Explain why these ratios are equivalent.

When Han makes chocolate milk, he mixes 2 cups of milk with 3 tablespoons of chocolate syrup. Here is a table that shows how to make batches of different sizes.

Use the information in the table to complete the statements.Some terms are used more than once.

1. The table shows a proportional relationship between ______________ and ______________.
2. The scale factor shown is ______________.
3. The constant of proportionality for this relationship is ______________.
4. The units for the constant of proportionality are ______________ per ______________.

Bank of Terms: tablespoons of chocolate syrup, $4$, cups of milk, cup of milk, $\frac32$

A certain shade of pink is created by adding 3 cups of red paint to 7 cups of white paint.

1. How many cups of red paint should be added to 1 cup of white paint?

   	cups of white paint	cups of red paint
   Row 1	1
   Row 2	7	3

2. What is the constant of proportionality?

A map of a rectangular park has a length of 4 inches and a width of 6 inches. Ituses a scale of 1 inch for every 30 miles.

1. What is the actual area of the park? Show how you know.

2. The map needs to be reproduced at a different scale so that it has an area of 6 square inches and can fit in a brochure. At what scale should the mapbe reproduced so that it fits on the brochure? Show your reasoning.

Noah drew a scaled copy of Polygon P and labeled it Polygon Q.

If the area of Polygon P is 5 square units, what scale factor did Noah apply to Polygon P to create Polygon Q? Explain or show how you know.

Select all the ratios that are equivalent to each other.

1. $4:7$
2. $8:15$
3. $16:28$
4. $2:3$
5. $20:35$

Noahis running a portion of a marathon at a constant speed of 6 miles per hour.

One kilometer is 1000 meters.

1. Complete the tables. What is the interpretation of the constant of proportionality in each case?

   row 1	meters	kilometers
   row 2	1,000	1
   row 3	250
   row 4	12
   row 5	1

   row 1	kilometers	meters
   row 2	1	1,000
   row 3	5
   row 4	20
   row 5	0.3

   The constant of proportionality tells us that:

   The constant of proportionality tells us that:

2. What is the relationship between the two constants of proportionality?

Which of these scales is equivalent to the scale 1 cm to 5 km? Select all that apply.

1. 3 cm to 15 km

2. 1 mm to 150 km

3. 5 cm to 1 km

4. 5 mm to 2.5 km

5. 1 mm to 500 m

Which one of these pictures is not like the others? Explain what makes it different using ratios.

A certain ceiling is made up of tiles. Every square meter of ceiling requires 10.75 tiles. Fill in the table with the missing values.

Each table represents a proportional relationship. For each, find the constant of proportionality, and write an equation that represents the relationship.

Here is a polygon on a grid.

1. Draw a scaled copy of the polygon using a scale factor 3. Label the copy A.

2. Draw a scaled copy of the polygon with a scale factor $\frac{1}{2}$. Label it B.

3. Is Polygon A a scaled copy of Polygon B? If so, what is the scale factor that takes B to A?

The table represents the relationship between a length measured in meters and the same length measured in kilometers.

A store sells rope by the meter. The equation $p = 0.8L$ represents the price $p$ (in dollars) of a piece of nylon rope that is $L$ meters long.

1. How much does the nylon rope cost per meter?
2. How long is a piece of nylon rope that costs $1.00?

The table represents a proportional relationship. Find the constant of proportionalityand write an equation to represent the relationship.

Constant of proportionality: __________

Equation: $y =$

On a map of Chicago, 1 cm represents 100 m. Select all statements that express the same scale.

1. 5 cm on the map represents 50 m in Chicago.

2. 1 mm on the map represents 10 m in Chicago.

3. 1 km in Chicago is represented by 10 cm the map.

4. 100 cm in Chicago is represented by 1 m on the map.

A car is traveling down a highway at a constant speed, described by the equation $d = 65t$, where$d$ represents the distance, in miles, that the car travels at this speed in $t$ hours.

1. What does the 65 tell us in this situation?
2. How many miles does the car travel in 1.5 hours?
3. How long does it take the car to travel 26 miles at this speed?

Elena has some bottles of water that each holds 17 fluid ounces.

1. Write an equation that relates the number of bottles of water ($b$) to the total volume of water ($w$) in fluid ounces.
2. How much water is in 51 bottles?
3. How many bottles does it take to hold 51 fluid ounces of water?

There are about 1.61 kilometers in 1 mile. Let $x$ represent a distance measured in kilometers and $y$ represent the same distance measured in miles. Write two equations that relate a distance measured in kilometers and the same distance measured inmiles.

In Canadian coins, 16quarters is equal in value to 2 toonies.

1. Fill in the table.
2. What does the value next to 1 mean in this situation?

Each table represents a proportional relationship. For each table:

1. Fill in the missing parts of the table.
2. Draw a circle around the constant of proportionality.

Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would the constant of proportionality be?

A taxi service charges \$1.00 for the first $\frac{1}{10}$ milethen \$0.10 for each additional $\frac{1}{10}$ mile after that.

A rabbit and turtle are in a race. Is the relationship between distance traveled and time proportional for either one? If so, write an equation that represents the relationship.

For each table, answer: What is the constant of proportionality?

The relationship between a distance in yards ($y$) and the same distance in miles ($m$) is described by the equation $y = 1760m$.

1. Find measurements in yards and miles for distances by filling in the table.

   row 1	distance measured in miles	distance measured in yards
   row 2	1
   row 3	5
   row 4		3,520
   row 5		17,600

2. Is there a proportional relationship between a measurement in yards and a measurement in miles for the same distance? Explain why or why not.

Decide whether or not each equation represents a proportional relationship.

1. The remaining length ($L$) of 120-inch rope after $x$ inches have been cut off:$120-x = L$
2. The total cost ($t$) after 8% sales tax is added to an item's price ($p$):$1.08p = t$
3. The number of marbles each sister gets ($x$) when $m$ marbles are shared equally among four sisters:$x = \frac{m}{4}$
4. The volume ($V$) of a rectangular prism whose height is 12 cm and base is a square with side lengths $s$ cm:$V = 12s^2$

1. Use the equation $y = \frac52 x$ to fill in the table.

   Is $y$ proportional to $x$ and $y$?Explain why or why not.

   	$x$	$y$
   row 1	2
   row 2	3
   row 3	6

2. Use the equation $y = 3.2x+5$ to fill in the table.

   Is $y$ proportional to $x$ and $y$? Explain why or why not.

   	$x$	$y$
   row 1	1
   row 2	2
   row 3	4

To transmit information on the internet, large files are broken into packets of smaller sizes. Each packet has 1,500 bytes of information. An equation relating packets to bytes of information is given by $b= 1,\!500p$ where $p$ represents the number of packets and $b$ represents the number of bytes of information.

1. How many packets would be needed to transmit 30,000 bytes of information?
2. How much information could be transmitted in 30,000 packets?
3. Each byte contains 8 bits of information. Write an equation to represent the relationship between the number of packets and the number of bits.

For each situation, explain whether you think the relationship is proportional or not. Explain your reasoning.

Every package of a certain toy also includes2 batteries.

1. Are the number of toys and number of batteries in a proportional relationship? If so, what are the two constants of proportionality? If not, explain your reasoning.
2. Use $t$ for the number of toys and $b$ for the number of batteries to write two equations relating the two variables.

   $b= $

   $t= $

Lin and her brother were born on the same date in different years. Lin was 5 years old when her brother was 2.

1. Find their ages in different years by filling in the table.

   row 1	Lin's age	Her brother's age
   row 2	5	2
   row 3	6
   row 4	15
   row 5		25

2. Is there a proportional relationship between Lin’s age and her brother’s age? Explain your reasoning.

Quadrilateral A has side lengths 3, 4, 5, and 6. Quadrilateral B is a scaled copy of Quadrilateral A with a scale factor of 2. Select all of the following that are side lengths of Quadrilateral B.

1. 5
2. 6
3. 7
4. 8
5. 9

Which graphs could represent a proportional relationship? Explain how you decided.

A lemonade recipe calls for $\frac14$ cup of lemon juice for every cup of water.

Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would be the constant of proportionality?

1. The sizes you can print a photo

   row 1	width of photo (inches)	height of photo (inches)
   row 2	2	3
   row 3	4	6
   row 4	5	7
   row 5	8	10

2. The distance from which a lighthouse is visible.

   row 1	height of a lighthouse (feet)	distance it can be seen (miles)
   row 2	20	6
   row 3	45	9
   row 4	70	11
   row 5	95	13
   row 6	150	16

Select all of the pieces of information that would tell you $x$ and $y$ have a proportional relationship. Let $y$ representthe distance between a rock and a turtle's current position in meters and $x$ representthe number of minutes the turtle has been moving.

1. $y = 3x$
2. After 4 minutes, the turtle has walked 12 feet away from the rock.
3. The turtle walks for a bit, then stops for a minute before walkingagain.
4. The turtle walks away from the rock at a constant rate.

There is a proportional relationship between the number ofmonths a person has had a streaming movie subscription and the total amount of money they have paid for the subscription. The cost for 6 months is \$47.94. The point $(6, 47.94)$ is shown on the graph below.

1. What is the constant of proportionality in this relationship?
2. What does the constant of proportionality tell us about the situation?
3. Add at least three more points to the graph and label them with their coordinates.
4. Write an equation that represents the relationship between $C$, the total cost of the subscription, and $m$, the number of months.

The graph shows the amounts of almonds, in grams,for different amounts of oats, in cups,in a granola mix. Label the point $(1, k)$ on the graph, find the value of $k$, and explain its meaning.

To make a friendship bracelet, some long strings are lined up thentaking one string and tying it in a knot with each of the other strings to create a row of knots. A new string is chosen and knotted with the all the other strings to create a second row. This process is repeated until there are enough rows to make a bracelet to fit around your friend's wrist.

Are the number of knotsproportional to the number of rows? Explain your reasoning.

Match each equation to its graph.

The graphs below show some data from a coffee shop menu. One of the graphs shows cost (in dollars) vs. drink volume (in ounces), and one of the graphs shows calories vs. drink volume (in ounces).

1. Which graph is which? Give them the correct titles.
2. Which quantities appear to be in a proportional relationship? Explain how you know.
3. For the proportional relationship, find the constant of proportionality. What does that number mean?

Lin and Andre bikedhome from school at a steady pace. Lin biked 1.5 km and it took her 5 minutes. Andre biked 2 km and it took him 8minutes.

1. Draw a graph with two lines that representthe bike rides of Lin and Andre.
2. For each line, highlight the point with coordinates $(1,k)$ and find $k$.
3. Who was biking faster?

At the supermarket you can fill your own honey bear container. A customer buys12oz of honey for \$5.40.

4. Choose one of your equations, and sketch its graph. Be sure to label the axes.

The point $(3, \frac65)$ lies on the graph representing a proportional relationship. Which of the followingpoints also lie on the same graph? Selectallthat apply.

A trail mix recipe asks for 4 cups of raisins for every 6 cups of peanuts. There is proportional relationship between the amount of raisins, $r$ (cups), and the amount of peanuts, $p$ (cups),in this recipe.

1. Write the equation for the relationship that has constant of proportionality greater than 1. Graph the relationship.
2. Write the equation for the relationship that has constant of proportionality less than 1. Graph the relationship.

Here is a graph that represents a proportional relationship.

The equation $c = 2.95g$ shows how much it costs to buy gas at a gas station on a certain day. In the equation, $c$ represents the cost in dollars, and $g$ represents how many gallons of gas were purchased.

1. Write down at least four (gallons of gas, cost) pairs that fit this relationship.
2. Create a graph of the relationship.
3. What does 2.95 represent in this situation?
4. Jada’s mom remarks, “You can get about a third of a gallon of gas for a dollar.” Is she correct? How did she come up with that?

There is a proportional relationship between a volume measured in cups and the same volume measured in tablespoons. 3 cups is equivalent to 48 tablespoons, as shown in the graph.

1. Plot and label at least two more points that represent the relationship.
2. Use a straightedge to draw a line that represents this proportional relationship.
3. For which value y is ($1, y$) on the line you just drew?
4. What is the constant of proportionality for this relationship?
5. Write an equation representing this relationship. Use $c$ for cups and $t$ for tablespoons.