This handy Spectrum Math Grade 7 Answer Key Chapter 4 Lesson 4.4 Using Equations to Represent Proportion provides detailed answers for the workbook questions.
Spectrum Math Grade 7 Chapter 4 Lesson 4.4 Using Equations to Represent Proportion Answers Key
Sometimes words are used to describe the proportional relationship in a problem. The words can tell how to write an equation to represent a proportional relationship.
A handicapped-access ramp starts at ground level and rises to 27 inches over a distance of 30 feet. What is the equation to find the height of the ramp based on how far along the ramp you have traveled?
1. Use the equation to find the constant of proportionality: k = \(\frac{y}{x}\). For simplicity, use the known value (27) of the variable you will be solving for (height) as y when setting up the proportion.
2. In this problem, k = \(\frac{27}{30}\), where 27 is the height of the ramp (y) and 30 is the distance it covers (x).
3. Simplify to k = \(\frac{9}{10}\). This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{9}{10}\) = \(\frac{y}{x}\).
4. With this proportion, you can find the height at any point along the ramp. Just isolate the variable you are solving for (x, or height) on one side of the equation. So, y = \(\frac{9}{10}\) × x.
Write the equation to solve each problem. Use y as the variable you solve for.
Question 1.
A recipe to make 4 pancakes calls for 6 tablespoons of flour. Tracy wants to make 10 pancakes using this recipe. What equation will she need to use to find out how many tablespoons of flour to use?
Equation: __________
Answer: Equation: f = \(\frac{6}{4}\) × p
A recipe to make 4 pancakes calls for 6 tablespoons of flour.
Tracy wants to make 10 pancakes using this recipe.
1. Use the equation to find the constant of proportionality: k = \(\frac{p}{f}\).
where p denotes number of pancakes and f represents number of table spoons of flour
2. In this problem, k = \(\frac{4}{6}\), where 4 represents number of pancakes and 6 represents number of table spoons of flour
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{4}{6}\) = \(\frac{p}{f}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as f = \(\frac{6}{4}\) × p
by simplification, f = \(\frac{3}{2}\) × p
As Tracy wants to make 10 pancakes, substitute 10 for p in the above equation
So, f =\(\frac{3}{2}\) × p = \(\frac{3}{2}\) × 10 = 15 tablespoons of flour
Question 2.
A picture measures 11 inches tall by 14 inches wide. Nathan wants to enlarge the picture to fit in a frame that is 16 inches wide. What equation will he need to use to find out how tall the picture should be after it is enlarged?
Equation: __________
Answer: Equation: t = \(\frac{11}{14}\) × w
A picture measures 11 inches tall by 14 inches wide.
Nathan wants to enlarge the picture to fit in a frame that is 16 inches wide.
1. Use the equation to find the constant of proportionality: k = \(\frac{w}{t}\).
where w represents width of the picture and t represents length of the picture
2. In this problem, k = \(\frac{14}{11}\), where 14 represents width of the picture and 11 represents length of the picture
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{14}{11}\) = \(\frac{w}{t}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as t = \(\frac{11}{14}\) × w
As Nathan wants to enlarge the picture to fit in a frame that is 16 inches wide, substitute 16 for t in the above equation.
t = \(\frac{11}{14}\) × w = \(\frac{11}{14}\) × 16 = \(\frac{176}{14}\) = 12.571 inches tall
Question 3.
A car uses 8 gallons of gasoline to travel 290 miles. Juanita wants to take a trip that is 400 miles. What equation will she need to use to find out how much gas the trip will use?
Equation: _____________
Answer: Equation: g = \(\frac{8}{290}\) × m
A car uses 8 gallons of gasoline to travel 290 miles.
Juanita wants to take a trip that is 400 miles.
1. Use the equation to find the constant of proportionality: k = \(\frac{m}{g}\).
where m represents number of miles and g represents amount of gasoline required.
2. In this problem, k = \(\frac{290}{8}\), where 290 represents number of miles and 8 represents amount of gasoline required.
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{290}{8}\) = \(\frac{m}{g}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as g = \(\frac{8}{290}\) × m
As Juanita wants to take a trip that is 400 miles, substitute 400 for m in the above equation
g = \(\frac{8}{290}\) × m = \(\frac{8}{290}\) × 400 = 11.034 gallons of gasoline
Question 4.
After Marco has worked for 5 hours, he has earned $29.00. He is planning to work 30 hours this week. What equation will he need to use to find out how much he will be paid?
Equation: _____________
Answer: Equation: e = \(\frac{29}{5}\) × w
After Marco has worked for 5 hours, he has earned $29.00.
He is planning to work 30 hours this week.
1. Use the equation to find the constant of proportionality: k = \(\frac{e}{w}\).
where e represents his earnings and w represents number of hours he worked
2. In this problem, k = \(\frac{29}{5}\), where 29 represents his earnings and 5 represents number of hours he worked
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{29}{5}\) = \(\frac{e}{w}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as e = \(\frac{29}{5}\) × w
As He is planning to work 30 hours this week, substitute 30 for w in the above equation
e = \(\frac{29}{5}\) × w = \(\frac{29}{5}\) × 30 = $174.00
Write the equation to solve each problem. Use x as the variable you solve for.
Question 1.
Chester wants to plant a flower bed that is 80 square feet. Each packet of seeds gives him enough flowers to cover 10 square feet of the flower bed. What equation will he use to find out how many packets of seeds to buy for his flower bed?
Equation: __________
Answer: Equation: x = \(\frac{1}{10}\) × a = \(\frac{1}{10}\) × 80
Chester wants to plant a flower bed that is 80 square feet.
Each packet of seeds gives him enough flowers to cover 10 square feet of the flower bed.
1. Use the equation to find the constant of proportionality: k = \(\frac{x}{a}\).
where x represents number of packets of seeds and a represents the area in square feet
2. In this problem, k = \(\frac{1}{10}\), where 1 represents number of packets of seeds and 10 represents the area in square feet
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{1}{10}\) = \(\frac{x}{a}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as x = \(\frac{1}{10}\) × a
As Chester wants to plant a flower bed that is 80 square feet, substitute 80 in place of a
x = \(\frac{1}{10}\) × a = \(\frac{1}{10}\) × 80 = 8 packets of seeds
Question 2.
At the Charming Chair Factory, they make 20 chairs per day when 5 workers are on duty. If they need to make 100 chairs in one day, what equation should they use to figure out how many workers to schedule?
Equation: __________
Answer: Equation: x = \(\frac{5}{20}\) × c = \(\frac{5}{20}\) × 100
At the Charming Chair Factory, they make 20 chairs per day when 5 workers are on duty.
they need to make 100 chairs in one day
1. Use the equation to find the constant of proportionality: k = \(\frac{x}{c}\).
where x represents number of workers and c represents number of chairs
2. In this problem, k = \(\frac{5}{20}\), where 5 represents number of workers and 20 represents number of chairs
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{5}{20}\) = \(\frac{x}{c}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as x = \(\frac{5}{20}\) × c
As they need to make 100 chairs in one day, substitute 100 in the place of c
Therefore, x = \(\frac{5}{20}\) × c = \(\frac{5}{20}\) × 100 = 25 workers
Question 3.
Henry is an artist who can produce 5 paintings every 2 months. He is getting ready for an exhibit and has to make 8 new paintings. What equation should he use to figure out how long it will take him to get his paintings ready?
Equation: __________
Answer: Equation: x = \(\frac{2}{5}\) × p = \(\frac{2}{5}\) × 8
Henry is an artist who can produce 5 paintings every 2 months.
He is getting ready for an exhibit and has to make 8 new paintings.
1. Use the equation to find the constant of proportionality: k = \(\frac{x}{p}\).
where x represents number of months and p represents number of paintings
2. In this problem, k = \(\frac{2}{5}\), where 2 represents number of months and 5 represents number of paintings
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{2}{5}\) = \(\frac{x}{p}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as x = \(\frac{2}{5}\) × p
As He is getting ready for an exhibit and has to make 8 new paintings, subtitute8 in the place of p
Therefore, x = \(\frac{2}{5}\) × p = \(\frac{2}{5}\) × 8 = \(\frac{16}{5}\) = 3\(\frac{1}{5}\) months
Question 4.
Andrea rents a bike for 8 hours and pays $42.00 for the rental. Tomorrow, she wants to rent the same bike, but only needs it for 6 hours. What equation can she use to figure out how much she will need to pay?
Equation: _____________
Answer: Equation: x = \(\frac{42}{8}\) × h = \(\frac{42}{8}\) × 6
Andrea rents a bike for 8 hours and pays $42.00 for the rental.
Tomorrow, she wants to rent the same bike, but only needs it for 6 hours.
1. Use the equation to find the constant of proportionality: k = \(\frac{x}{h}\).
where x represents amount she needs to pay and h represents number of hours
2. In this problem, k = \(\frac{42}{8}\), where 42 represents amount she needs to pay and 8 represents number of hours
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{42}{8}\) = \(\frac{x}{h}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as x = \(\frac{42}{8}\) × h
As she wants to rent the same bike tomorrow, but only needs it for 6 hours, substitute 6 in the place of h
Therefore, x = \(\frac{42}{8}\) × h = \(\frac{42}{8}\) × 6 = $31.50
Question 5.
Sara can bake 12 cookies with 2 scoops of flour. If she wants to make 36 cookies, what equation should she use to help her find out how many scoops of flour to use?
Equation: __________
Answer: Equation: x = \(\frac{2}{12}\) × c = \(\frac{2}{12}\) × 36
Sara can bake 12 cookies with 2 scoops of flour.
she wants to make 36 cookies
1. Use the equation to find the constant of proportionality: k = \(\frac{x}{c}\).
where x represents number of scoops of flour and c represents number of cookies
2. In this problem, k = \(\frac{2}{12}\), where 2 represents number of scoops of flour and12 represents number of cookies
3. This is the constant of proportionality for this problem, so you can plug this value into the equation in step 1 to get \(\frac{2}{12}\) = \(\frac{x}{c}\).
4. With this proportion, you can find the number of tablespoons of flour to use.
Therefore, the equation can be written as x = \(\frac{2}{12}\) × c
As she wants to make 36 cookies, substitute 36 in the place of c
Therefore, x = \(\frac{2}{12}\) × c = \(\frac{2}{12}\) × 36 = 6 scoops of flour