## Engage NY Eureka Math Grade 6 Module 4 Lesson 28 Answer Key

### Eureka Math Grade 6 Module 4 Lesson 28 Mathematical Modeling Exercise Answer Key

Mathematical Modeling Exercise:

Question 1.
Juan has gained 20 lb. since last year. He now weighs 120 lb. Rashod is 15 Ib. heavier than Diego. If Rashod and Juan weighed the same amount last year, how much does Diego weigh? Let j represent Juanâ€™s weight last year in pounds, and let d represent Diegoâ€™s weight in pounds.

Draw a tape diagram to represent Juanâ€™s weight.

Draw a tape diagram to represent Rashodâ€™s weight.

Draw a tape diagram to represent Diegoâ€™s weight.

What would combining all three tape diagrams look like?

Write an equation to represent Juanâ€™s tape diagram.
j + 20 = 120

Write an equation to represent Rashodâ€™s tape diagram.
d + 15 + 20 – 120

How can we use the final tape diagram or the equations above to answer the question presented?
By combining 15 and 20 from Rashodâ€™s equation, we can use our knowledge of addition identities to determine Diegoâ€™s weight.
The final tape diagram can be used to write a third equation d + 35 = 120. We con use our knowledge of addition identities to determine Diegoâ€™s weight
Calculate Diegoâ€™s weight.
d + 35 – 35 – 120 – 35
d = 85
We can use identities to defend our thought that d + 35 – 35 = d.

Yes. If Diego weighs 85 lb., and Rashod weighs 15 lb. more than Diego, then Rashod weighs 100 lb., which is what Juan weighed before he gained 20 lb.

### Eureka Math Grade 6 Module 4 Lesson 28 Example Answer Key

Example 1:

Marissa has twice as much money as Frank. Christina has $20 more than Marissa. If Christina has$100, how much money does Frank have? Let f represent the amount of money Frank has in dollars and m represents the amount of money Marissa has in dollars.

Draw a tape diagram to represent the amount of money Frank has.

Draw a tape diagram to represent the amount of money Marissa has.

Draw a tape diagram to represent the amount of money Christina has.

Which tape diagram provides enough information to determine the value of the variable m?
The tape diagram represents the amount of money Christina has.

Write and solve the equation.
m + 20 = 100
m + 20 – 20 = 100 – 20
m = 80
The identities we have discussed throughout the module solidify that m + 20 – 20 = m

What does the 80 represent?
80 is the amount of money, in dollars, that Morissa has.

Now that we know Marissa has $80, how can we use this information to find out how much money Frank has? Answer: We can write an equation to represent Marissaâ€™s tape diagram since we now know the length is 80. Write an equation. Answer: 2f = 80 Solve the equation. Answer: 2f Ã· 2 = 80 – 2 f = 40 Once again, the identities we have used throughout the module can solidify that 2f Ã· 2 = f. What does the 40 represent? Answer: The 40 represents the amount of money Frank has in dollars. Does 40 make sense in the problem? Answer: Yes, because if Frank has$40, then Marissa has twice this, which is $80. Then, Christina has$100 because she has \$20 more than Marissa, which is what the problem stated.

### Eureka Math Grade 6 Module 4 Lesson 28 Exercise Answer Key

Station One: Use tape diagrams to solve the problem:

Raeana is twice as old as Madeline, and Laura is 10 years older than Raeana. If Laura is 50 years old, how old is Madeline? Let m represent Madelineâ€™s age in years, and let r represent Raeanaâ€™s age in years.

Raeanoâ€™s Tape Diagram:

Lauraâ€™s Tape Diagram:

Equation for Lauraâ€™s Tape Diagram:
r Ã· 10 = 50
r + 10 – 10 = 50 – 10
r = 40
We now know that Raeana is 40 years old, and we can use this and Raeanaâ€™s tape diagram to determine the age of
2m = 40
2m – 2 = 40 Ã· 2
m = 20
There are, Madeline is 20 years old.

Station Two: Use tape diagrams to solve the problem.

Cadi has 90 apps on her phone. Braylen has half the amount of apps as Thees. If Cadi has three times the amount of apps as Theiss how many apps does Braylen have? Let b represent the number of Braylenâ€™s apps and t represent the number of Theissâ€™s apps.
Theissâ€™s Tape Diagram:

Broylenâ€™s Tape Diagram:

Carliâ€™s Tape Diagram:

Equation for Corllâ€™s Tape Dio gram:
3t = 90
3t Ã· 3 = 90 + 3
t = 30
We now know that fleas has 30 opps on his phone. We can use this information to write an equation for Braylenâ€™s tape diagram and determine how many opps ore on Bra yknâ€™s phone.
2h = 30
2b Ã· 2 = 30 + 2
b = 15

Therefore, Broylen has 15 apps on his phone.

Station Three: Use tape diagrams to solve the problem.

Reggie ran for 180 yards during the last football game, which is 40 more yards than his previous personal best Monte ran 50 more yards than Adrian during the same game. If Monte ran the same amount of yards Reggie ran in one game for his previous personal best, how many yards did Adrian run? Let r represent the number of yards Reggie ran during his previous personal best and a represents the number of yards Adrian ran.
Reggieâ€™s Tape Diagram:

Monte’s Tape Diagram:

Combining all 3 tape diagrams:

Equation for Reggieâ€™s Tape Diagram:
r + 40 = 180

Equation for Monteâ€™s Tape Diagram:
a + 50 + 40 = 180
a + 90 = 180
a + 90 – 90 = 180 – 90
a = 90
Therefore, Adrian ran 90 yards during the football game.

Station Four: Use tape diagrams to solve the problem.

Lance rides his bike downhill at a pace of 60 miles per hour. When Lance is riding uphill, he rides 8 miles per hour slower than on flat roads. If Lanceâ€™s downhill speed is 4 times faster than his flat-road speed, how fast does he travel uphill? Let f represent Lanceâ€™s pace on flat roads in miles per hour and u represent Lanceâ€™s pace uphill in miles per hour.
Tape Diagram for Uphill Pace:

Tape Diagram for Downhill Pace:

Equation for Downhill Pace:
4f = 60
4f Ã· 4 = 60 Ã·4
f = 15

Equation for Uphill Pace:
u + 8 = 15
u + 8 – 8 = 15 – 8
u = 7
Therefore, Lance travels at a pace of 7 miles per hour uphill.

### Eureka Math Grade 6 Module 4 Lesson 28 Problem Set Answer Key

Use tape diagrams to solve each problem.

Question 1.
Dwayne scored SS points in the last basketball game, which is 10 points more than his previous personal best. Lebron scored 15 points more than Chris in the same game. Lebron scored the same number of points as Dwayneâ€™s previous personal best. Let d represent the number of points Dwayne scored during his previous personal best and c represent the number of Chrisâ€™s points.

a. How many points did Chris score during the game?

Equation for Dwayneâ€™s Tape Diagram: d + 10 = 55
Equation for Lebrons Tape Diagram:
c + 15 + 10 = 55
c + 25 = 55
c + 25 – 25 = 55 – 25
c = 30
Therefore, Chris scored 30 points in the game.

b. If these are the only three players who scored, what was the teamâ€™s total number of points at the end of the game?
Dwayne scored 55 points. Chris scored 30 points. Lebron scored 45 points (answer to Dwayreâ€™s equation). Therefore, the total number of points scored is 55 + 30 + 45 = 130.

Question 2.
The number of customers at Yummy Smoothies varies throughout the day. During the lunch rush on Saturday, there were 120 customers at Yummy Smoothies. rhe number of customers at Yummy Smoothies during dinner time was 10 customers fewer than the number during breakfast. The number of customers at Yummy Smoothies during lunch was 3 times more than during breakfast. How many people were at Yummy Smoothies during breakfast? How many people were at Yummy Smoothies during dinner? Let d represent the number of customers at Yummy Smoothies during dinner and b represent the number of customers at Yummy Smoothies during breakfast.
Tape Diagram for Lunch:

Tape Diagram for Dinner:

Equation for Lunchs Tape Diagram:
3b = 120
3b Ã· 3 = 120 Ã· 3
b = 40

Now that we know 40 customers were at Yummy Smoothies for breakfast, we can use this information and the tape diagram for dinner to determine how many customers were at Yummy Smoot hies during dinner.
d + 10 = 40
d + 10 – 10 = 40 – 10
d = 30
Therefore, 30 customers were at Yummy Smoothies during dinner and 40 customers during breakfast.

Question 3.
Karter has 24 T-shirts. Karter has 8 fewer pairs of shoes than pairs of pants. If the number of T-shirts Karter has is double the number of pants he has, how many pairs of shoes does Karter have? Let p represent the number of pants Karter has and s represent the number of pairs of shoes he has.
Tape Diagram for T-shirts:

Tape Diagram for Shoes:

Equation for T-Shirts Tape Diagram:
2p = 24
2p Ã· 2 = 24 Ã· 2
p = 12

Equation for Shoes Tape Diagram:
s + 8 = 12
s + 8 – 8 = 12 – 8
s = 4
Karter has 4 pairs of shoes.

Question 4.
Darnell completed 35 push-ups in one minute, which is 8 more than his previous personal best. Mia completed 6 more push-ups than Katie. If Mia completed the same amount of push-ups as Darnell completed during his previous personal best, how many push-ups did Katie complete? Let d represent the number of push-ups Darnell completed during his previous personal best and k represent the number of push-ups Katie completed.

d + 8 = 35
k + 6 + 8 = 35
k + 14 = 35
k + 14 – 14 = 35 – 14
k = 21
Katie completed 21 push-ups.

Question 5.
Justine swims freestyle at a pace of 150 laps per hour, Justine swims breaststroke 20 laps per hour slower than she swims butterfly. If Justineâ€™s freestyle speed is three times faster than her butterfly speed, how fast does she swim breaststroke? Let b represent Justineâ€™s butterfly sped in laps per hour and r represent Justineâ€™s breaststroke sped in laps per hour.
Tape Diagram for breaststroke:

3b = 150
3b Ã· 3 = 150 Ã· 3
b = 50
Therefore, Justine swims butterfly at apace of 50 laps per hour.
r + 20 = 50
r + 20 – 20 = 50 – 20
r = 30
Therefore, Justine swims breaststroke at a pace of 30 laps per hour.

### Eureka Math Grade 6 Module 4 Lesson 28 Exit Ticket Answer Key

Use tape diagrams and equations to solve the problem with visual models and algebraic methods.

Question 1.
Alyssa is twice as old as Brittany, and Jazmyn is 15 years older than Alyssa. If Jazmyn is 35 years old, how old is Brittany? Let a represent Alyssaâ€™s age in years and b represent Brittanyâ€™s age in years.
Brittanyâ€™s Tape Diagram:

Alyssaâ€™s Tape Diagram:

Jazmynâ€™s Tape Diagram:

Equation for jazmynâ€™s Tape Diagram:
a + 15 = 35
a + 15 – 15 = 35 – 15
a = 20

Now that we know Alyssa is 20 years old, we can use this information and Alyssaâ€™s tape diagram to determine Brittanyâ€™s age.
2b = 20
2b Ã· 2 = 20 Ã· 2
b = 10
Therefore, Brittany is 10 years old.

Addition of Decimals II – Round 1:

Directions: Evaluate each expression:

Question 1.
2.5 + 4
6.5

Question 2.
2.5 + 0.4
2.9

Question 3.
2.5 + 0.04
2.54

Question 4.
2.5 + 0.004
2.504

Question 5.
2.5 + 0.0004
2.5004

Question 6.
6 + 1.3
7.3

Question 7.
0.6 + 1.3
1.9

Question 8.
0.06 + 1.3
1.36

Question 9.
0.006 + 1.3
1.306

Question 10.
0.0006 + 1.3
1.3006

Question 11.
0.6 + 13
13.6

Question 12.
7 + 0.2
7.2

Question 13.
0.7 + 0.02
0.72

Question 14.
0.07 + 0.2
0.27

Question 15.
0.7 + 2
2.7

Question 16.
7 + 0.02
7.02

Question 17.
6 + 0.3
6.3

Question 18.
0.6 + 0.03
0.63

Question 19.
0.06 + 0.3
0.36

Question 20.
0.6 + 3
3.6

Question 21.
6 + 0.03
6.03

Question 22.
0.6 + 0.3
0.9

Question 23.
4.5 + 3.1
7.6

Question 24.
4.5 + 0.31
4.81

Question 25.
4.5 + 0.031
4.531

Question 26.
0.45 + 0.031
0.481

Question 27.
0.045 + 0.03 1
0.076

Question 28.
12 +0.36
12.36

Question 29.
1.2 + 3.6
4.8

Question 30.
1.2 + 0.36
1.56

Question 31.
1.2 + 0.036
1.236

Question 32.
0.12 + 0.036
0.156

Question 33.
0.012 + 0.036
0.048

Question 34.
0.7 + 3
3.7

Question 35.
0.7 + 0.3
1

Question 36.
0.07 + 0.03
0.1

Question 37.
0.007 + 0.003
0.01

Question 38.
5 + 0.5
5.5

Question 39.
0.5 + 0.5
1

Question 40.
0.05 + 0.05
0.1

Question 41.
0.005 + 0.005
0.01

Question 42.
0.11+ 19
19.11

Question 43.
1.1 + 1.9
3

Question 44.
0.11+0.19
0.3

Addition of Decimals II – Round 2:

Directions: Evaluate each expression:

Question 1.
7.4 + 3
10.4

Question 2.
7.4 + 0.3
7.7

Question 3.
7.4 + 0.03
7.43

Question 4.
7.4 + 0.003
7.403

Question 5.
7.4 + 0.0003
7.4003

Question 6.
6 + 2.2
8.2

Question 7.
0.6 + 2.2
2.8

Question 8.
0.06 + 2.2
2.26

Question 9.
0.006 + 2.2
2.206

Question 10.
0.0006 + 2.2
2.2006

Question 11.
0.6 + 22
22.6

Question 12.
7 + 0.8
7.8

Question 13.
0.7 + 0.08
0.78

Question 14.
0.07 + 0.8
0.87

Question 15.
0.7 + 8
8.7

Question 16.
7 + 0.08
7.08

Question 17.
5 + 0.4
5.4

Question 18.
0.5 + 0.04
0.54

Question 19.
0.05 + 0.4
0.45

Question 20.
0.5 + 4
4.5

Question 21.
5 + 0.04
5.04

Question 22.
5 + 0.4
5.4

Question 23.
3.6 + 2.3
5.9

Question 24.
3.6 + 0.23
3.83

Question 25.
3.6 + 0.023
3.623

Question 26.
0.36 + 0.023
0.383

Question 27.
0.036 + 0.023
0.059

Question 28.
0.13 + 56
56.13

Question 29.
1.3 + 5.6
6.9

Question 30.
1.3 + 0.56
1.86

Question 31.
1.3 + 0.056
1.356

Question 32.
0.13 + 0.056
0.186

Question 33.
0.013 + 0.056
0.069

Question 34.
2 + 0.8
2.8

Question 35.
0.2 + 0.8
1

Question 36.
0.02 + 0.08
0.1

Question 37.
0.002 + 0.008
0.01

Question 38.
0.16 + 14
14.16

Question 39.
1.6 + 1.4
3

Question 40.
0.16 + 0.14
0.3

Question 41.
0.016+0.014
0.03

Question 42.
15 + 0.15