We included **HMH Into Math Grade 6 Answer Key PDF** **Module 10 Lesson 2 Write Equations from Verbal Descriptions **to make students experts in learning maths.

## HMH Into Math Grade 6 Module 10 Lesson 2 Answer Key Write Equations from Verbal Descriptions

I Can identify the dependent and independent variables, write equations to represent a real-world situation, and use the equations to solve problems.

**Step It Out**

Question 1.

Ms. Tran tutors students in mathematics. Her rate is shown. How much does Ms. Tran earn tutoring for any number of hours?

A. Identify the independent and dependent variables.

Let x represent the number of hours Ms. Tran tutors.

Let y represent the number of dollars per hour Ms. Tran earns for tutoring.

the independent variable.

is the dependent variable.

Answer:

Given that,

The tutoring rate is $25.50 per hour.

Let x represent the number of hours Ms. Tran tutors.

Let y represent the number of dollars per hour Ms. Tran earns for tutoring.

From the given data the equation is y = 25.5x.

The independent variable is a variable that is independent of the other value in the expression.

The dependent variable is a variable that depends on the other value in the expression.

The x is the independent variable.

The y is the dependent variable.

B. Write an equation to represent the situation.

The equation that represents the situation is ___________.

Answer:

Given that,

Ms. Tran’s total earnings in dollars = y

Number of hours spent tutoring = x

The equation that represents the situation is y = 25.5x

**Turn and Talk** How did you decide which variable is the independent variable?

Answer: When the independent variable causes the change in the dependent variable then you decide that the variable is the independent variable.

Question 2.

An orchestra will give a concert. There are 1,500 seats in the concert hall. Some tickets have been sold. How many tickets remain?

A. Identify the independent and dependent variables.

Let x represent the number of tickets sold.

Let y represent the number of tickets that remain.

is the independent variable.

is the dependent variable.

Answer:

Given that,

The total number of seats in the hall = 1500.

Let x represent the number of tickets sold.

Let y represent the number of tickets that remain.

From the given data the expression is y = 1500 – x.

The independent variable is a variable that is independent of the other value in the expression.

The dependent variable is a variable that depends on the other value in the expression.

x is the independent variable.

y is the dependent variable.

B. Write and solve an equation representing the situation.

The equation that represents the situation is ____________.

Answer:

Given that,

The total number of seats in the hall = 1500.

Let x represent the number of tickets sold.

Let y represent the number of tickets that remain.

The equation representing the situation is y = 1500 – x.

C. So far, 847 tickets have been sold. How many tickets remain?

Substitute the value for x, 847, into the equation. y = 1,500 –

Subtract to solve for y. y =

Given that 847 tickets to the concert have been sold, tickets remain.

Answer:

Given that x = 847.

The equation is y = 1500 – 847

y = 653.

Given that 847 tickets to the concert have been sold 653 tickets remain.

**Turn and Talk** Are there other equations that you can write to represent this situation? Explain.

Answer:

The other way for the above equation is y + 847 = 1500.

This equation also represents the above situation.

**Check Understanding**

Question 1.

Jim gives the cashier a $20 bill at the pharmacy. Write an equation that relates the change c he gets back in dollars to the cost b, in dollars, of his purchases. Which variable represents the independent variable? How much change does he get back if his purchases total $12.83?

Answer:

Given that,

The Jim’sgives the cashier a bill at the pharmacy = $20.

Jim’s purchases total = $12.83.

Therefore $20 – $12.83 = $7.17

The equation that relates c = change and b = back in dollars.

So, b = $12.83

c = $7.17

Jim’s receives $7.17.

Therefore c is the independent variable

Question 2.

A hot-air balloon is 45 meters above the top of the tallest building in a city. Write an equation to show the height b of the balloon if the tallest building is t meters tall.

Answer:

Given that,

The height of the building = t meters.

The height of the balloon = b

The hot-air balloon above the top of the tallest building in a city = 45 meters.

The equation is b = t + 45.

t is the independent variable.

b is the dependent variable.

**On Your Own**

Question 3.

Rosa earns $12.25 per hour working at an ice-cream shop.

A. Write an equation that shows Rosa’s earnings e, in dollars, for h hours of working at the ice-cream shop.

Answer:

Given that,

Rosa earns = $12.25 per hour.

Rosa’s earnings = e.

The working hours in the ice-cream shop = h.

The equation is e = $12.25h

h is the independent variable.

e is the dependent variable.

B. How much does Rosa earn for working 19\(\frac{1}{2}\) hours? Round your answer to the nearest cent.

Answer:

Given that.

19\(\frac {1}{2} \) = 19 x 1/2 hours

Rosa earns = $12.25 per hour.

Rosa’s earnings = e.

The working hours in the ice-cream shop = h = 19 x 1/2 hours.

The equation is e = $12.25(19 x 1/2).

e = $12.25(9.5)

e = $116.375

$116.375 rounded to the nearest cent is $116.

Question 4.

Lin is running a marathon.

A. Write an equation that shows the number of miles y she has left after running x miles.

Answer:

Given that,

A marathon is 26.2 miles long.

The number of miles = y miles

she has left after running = x miles

Therefore the equation is y = 26.2 – x

x is the independent variable.

y is the dependent variable.

B. If Lin has already run 19.6 miles of the marathon, how many miles does she have left?

Answer:

Given that,

A marathon is 26.2 miles long.

Lin has already run the marathon = 19.6

Here 26.2 – 19.6 = 6.6 miles.

Therefore Lin left 6.6 miles.

Question 5.

**Use Structure** You want to make 2\(\frac{1}{2}\) times as many muffins as your friend’s recipe makes. Write an equation that shows how to find the amount of each ingredient you will need for the adjusted recipe. Let x represent the amount of each ingredient in the original recipe and y represent the amount of each ingredient in the adjusted recipe.

Answer:

Given that,

2\(\frac {1}{2} \) = 2 x 1/2

You want to make as many muffins as your friend’s recipe makes

Let x represent the amount of each ingredient in the original recipe.

y represents the amount of each ingredient in the adjusted recipe.

Therefore the equation is y = x + 2 x 1/2.

x is the independent variable.

y is the dependent variable.

**Model with Mathematics For Problems 6-9, write an equation to represent each situation.**

Question 6.

Neil pays $19.99 for each shirt he buys. How many total dollars p does he pay for s shirts?

Answer:

Given that,

Neil pays for each shirt = $19.99

The number of shirts = s

Neil pays for s shirts = p.

p = $19.99 x s

s is the independent variable.

p is the dependent variable.

Question 7.

Omar has been practicing violin for 4\(\frac{1}{2}\) years longer than his sister. If his sister has been practicing for x years, how many years y has Omar been practicing violin?

Answer:

Given that

4\(\frac {1}{2} \) = 4.5 years.

Omar practiced violin longer than his sister = 4.5 years.

The total number of hours his sister practiced = x.

The total number of hours that Omar practiced = y.

The equation is y = x + 4.5.

x is the independent variable.

y is the dependent variable.

Question 8.

A coupon discounts the total grocery bill by $3.50. How many dollars g is the bill after the discount if the bill before the discount is b dollars?

Answer:

Given that,

The total Grocery bill = $3.50.

Bill after the discount = g dollars

Bill before the discount = b dollars

The equation is g = b – $3.50.

b is the independent variable.

g is the dependent variable.

Question 9.

Tickets for a local dog show cost $9.75 each. What is the total cost c, in dollars, for t tickets?

Answer:

Given that,

The cost of the tickets for the local dog show = $9.75 each.

The number of tickets = t.

The total cost for t tickets = c.

The equation is t = $9.75 x c.

c is the independent variable.

t is the dependent variable.

Question 10.

**STEM** Scientists measure temperature in degrees Celsius and in kelvin. A temperature in kelvin is 273.15 greater than the temperature in degrees Celsius.

A. Model with Mathematics What is an equation that relates the temperature K, in kelvin, to the temperature C, in degrees Celsius?

Answer:

Given that,

The temperature in kelvin is 273.15 greater than the temperature in degrees Celsius.

The temperature in kelvin = K

The temperature in degrees Celsius = C

Therefore the equation is C = K – 273.15.

C is the dependent variable.

K is the independent variable.

B. What is the temperature in kelvin of a gas that has a temperature of 5 °C?

Answer:

Given that,

The gas has a temperature of 5 °C.

The formula for the Celsius to kelvin conversion is K = °C + 273.15

K = 5 + 273.15

5 °C = 278.15 kelvin.

Question 11.

**Open Ended** Describe a scenario that matches the equation y = \(\frac{3}{2}\)x.

Answer:

Riya works \(\frac {3}{2} \) hours per each day. How many hours y does Riya work in x days?

Question 12.

A giraffe is 3 times as tall as a person. Write an equation for the height g, in meters, of a giraffe if a person is t meters tall. How tall is the giraffe if the person is 1.85 meters tall?

Answer:

Given that,

The giraffe is 3 times as long as the person.

The height of the giraffe = g.

The height of the person = h.

The equation is g = 3h

The height of the person = 1.85 meters.

The height of the giraffe is 1.85 x 3 = 5.55 meters.

Therefore the height of the giraffe is 5.55 meters.

Question 13.

Kayla walked 1,392 fewer steps than her mother walked Write an equation for the number of steps w Kayla walked if her mother walked s steps. If her mother walked 11,258 steps, how many steps did Kayla walk?

Answer:

Given that,

Kayla walked fewer steps than her mother walked = 1,392 steps.

Her mother walked = 11,258 steps

11,258 – 1392 = 9866

Kayla walked 9866 steps.

The number of steps that Kayla walked = w when her mother walked s steps.

Therefore the equation is w = 9866 + s.

w = dependent variable.

s = independent variable.

Question 14.

The diameter of a tree increases by about \(\frac{1}{8}\) inch each year. Write an equation that represents x, the increase in the diameter of the tree, in the last y years. How much has the diameter increased in the last 15 years?

Answer:

Given that,

\(\frac {1}{8} \) = 1/8 inches.

The diameter of the tree increases in each year = 1/8 inches

The equation is x = 1/8 x y

Y = 15

x = 1/8 x 15

x = 1.875 inches.

The diameter increased in the last 15 years is 1.875 inches.

Question 15.

**Model with Mathematics** Ramon wants to collect $250 in donations to support his favorite charity. He has collected f dollars from his friends and family but has not reached his goal. Write an equation that describes the number of dollars d Ramon still needs to collect to meet his goal. Which variable represents the independent variable?

Answer:

Ramon wants to collect the donations to support his favorite charity = $250.

He has collected = f.

The total number of dollars that Ramon collects = d.

Therefore the equation is d = $250 – f.

The variable f represents the independent variable.

Question 16.

Ms. Quinn pays $16.99 per person to take several friends to lunch. Write an equation that represents the cost C, in dollars, Ms. Quinn pays for f friends. How much does she pay for 7 friends?

Answer:

Given that,

Ms. Quinn pays per person = $16.99.

The total cost = C

Ms. Quinn pays for his friends = f

Therefore the equation is C = $16.99f.

f is the independent variable.

C is the dependent variable.

The number of friends = 7.

C = $16.99 x 7

= $118.93

She paid $118.93 to his 7 friends.

**Lesson 10.2 More Practice/Homework**

Question 1.

**Model with Mathematics** The area of a floor in square yards is one-ninth the area of the floor in square feet. Write the equation representing y, the area in square yards, to f, the area in square feet.

Answer:

Given that,

The area of a floor in square yards is = 1/9 the area of the floor in square feet.

The area in square yards = y.

The area in the square feet = f

Therefore the equation is y = 1/9 x f.

y = f/9

y is the dependent variable.

f is the independent variable.

Question 2.

An object’s height is measured in centimeters and inches. Write an equation that relates the number of centimeters c to the number of inches i. Determine the height in centimeters of a chair that is 32 inches tall.

Answer:

Given that,

The number of centimeters = c

The number of inches = I

Therefore the equation is c = 32i.

1 inch = 2.54cm

32 inches = 32 x 2.54

32 inches chair = 81.28 cm chair.

I is the independent variable.

c is the dependent variable.

Question 3.

**Model with Mathematics** Michael practiced his harmonica 3 times as long as Timothy did. Michael practiced for 126 minutes. Write an equation that represents the relationship between the amount of time Timothy practiced x and the amount of time Michael practiced.

Answer:

Given that,

Michael practiced his harmonica = 126 minutes.

Michael practiced his harmonica as long as Timothy did = 3 times.

The amount of time Timothy and Michael practiced = x

Therefore the equation is 126 = 3x

x is the dependent variable.

Question 4.

Caleb has read 17 more books than his friend Liana. Write an equation that represents the relationship between n, the number of books Caleb has read, and p. the number of books Liana has read.

Answer:

Given that,

Caleb has read more books than his friend Liana = 17

The number of books that Caleb read = n.

The number of books that Liana read = p.

Therefore the equation is n = p + 17.

p is the independent variable.

n is the dependent variable.

**Model with Mathematics For Problems 5-10, write an equation for each description.**

Question 5.

Felix sold 35 fewer raffle tickets than Helene. How many tickets F did Felix sell if Helene sold H tickets?

Answer:

Given that,

Felix sold fewer raffle tickets than Helene = 35.

The total tickets sold by Felix = F

The total tickets sold by Helene = H

Therefore the equation is F = H – 35.

H is the independent variable.

F is the dependent variable.

Question 6.

Sloane works 7\(\frac{1}{2}\) hours each day. How many hours h does Sloane work in d days?

Answer:

Given that,

7\(\frac {1}{2} \) = 7.5

Sloane works per day = 7.5 hours.

Number of days that the Sloane worked = d.

The number of hours Sloane works in d days = h.

Therefore the given equation is h = 7.5d.

d is the independent variable.

h is the dependent variable.

Question 7.

A grocery store charges $2.49 per pound for organic apples. What is the cost c, in dollars, of p pounds of organic apples?

Answer:

Given that,

The charge for the organic apples in the grocery store is = $2.49

The total number of pounds = p

The total charge for the p pounds of oranges apples = c.

Therefore the given equation is c = $2.49p.

p is the independent variable.

c is the dependent variable.

Question 8.

Paula used 1\(\frac{3}{4}\) cups of flour for a recipe. How many cups y of flour did Paula have if she had x cups before making the recipe?

Answer:

Given that,

1\(\frac {3}{4} \) = 1 x 3/4.

Paula used flour for a recipe = 1 x 3/4 cups.

The total number of cups before making the recipe = x

The total number of flour cups = y

Therefore the equation is y = x – 1 x 3/4.

x is the independent variable.

y is the dependent variable.

Question 9.

Each crate holds 135 avocados. How many avocados a are in c crates?

Answer:

Given that,

Each crate holds = 135 avocados.

Number of crates = c

Number of avocados = a

Therefore the equation is a = 135c.

c is the independent variable.

a is the dependent variable.

Question 10.

Every delivery order from a pizzeria has a delivery charge of $1.50. How many dollars b is the total bill for a delivery of one pizza if the pizza costs p dollars?

Answer:

Given that,

Every delivery order from a pizzeria has a delivery charge = $1.50

The cost of pizza = p

The total bill for the delivery of one pizza = b.

Therefore the equation is b = p + $1.50.

p is the independent variable.

b is the dependent variable.

**Test Prep**

Question 11.

Edith bought a science book and a math book for her college classes. The cost s, in dollars, of the science book is $28.75 more than the cost m, in dollars, of the math book.

A. Write an equation that relates the prices of the two books.

Answer:

Given that,

The cost of a math book = m

The cost of a science book = s – $28.75

The equation is m = s – $28.75

s is the independent variable.

m is the dependent variable.

B. How much does the math book cost if the science book cost $78.50?

Answer:

The equation is m = s – $28.75.

s = $78.50

Therefore $78.50 – $28.75

The cost of the math book is $49.75

Question 12.

A machine fills 75 bottles of water each minute. Write an equation to represent the number of bottles b of water the machine can fill in m minutes.

Answer:

Given that,

A machine fills the bottles of water for each minute = 75.

The total number of bottles = b

The bottles b of water the machine can fill in m minutes.

The equation is b = 75m

m is the independent variable.

b is the dependent variable

Question 13.

Ken said he would sell 30 tickets to the school play. Write an equation to relate the number of tickets the has left to sell to the number of tickets she has already sold. Which variable is the dependent variable?

Answer:

Given that,

Ken said he would sell tickets to the school play = 30.

The number of tickets she has left to sell = y

The number of tickets she has already sold = x

Therefore the equation is y = 30 – x.

x is the independent variable.

y is the dependent variable.

Question 14.

Viola is making bracelets to sell at a craft fair. Each bracelet will have 20 beads. Write an equation to relate the number of beads b she has to the number of bracelets x she can make.

Answer:

Given that,

The total number of beads in each bracelet = 20.

The total number of beads = b

The total number of bracelets = x.

The equation is b = 20x

x is the independent variable.

b is the dependent variable.

**Spiral Review**

Question 15.

Roberto paid $35.75 for dinner at a restaurant, plus an additional $7.15 tip. What percent of the bill did he pay as a tip? Round to the nearest percent, if necessary.

Answer:

Given that,

Roberto paid for the dinner = $35.75.

Roberto gave the tip = $7.15.

$7.15 what percent of $35.75 is

Let us consider = 100% = $35.75

x% = $7.15.

100%/x% = $35.75/$7.15

Multiply both sides of the equation with x

(100%/x%)x = ($35.75/$7.15)x

Then 100 = 5 × X

X = 100/5

X = 20%

Therefore 20% of the bill he paid has a tip.

20% rounded to the nearest percent is 20%.

Question 16.

Write an equation to represent the values in the table.

Answer:

The difference of x and y variables is

3 – 0 = 3

8 – 5 = 3

13 – 10 = 3

18 – 15 = 3

Therefore the equation is y = x – 3

x is the independent variable.

y is the dependent variable.

Question 17.

A 5-pound bag of organic brown rice costs $12.99. A 12-pound bag of the same type of rice costs $31.99. Which represents the better buy? Explain.

Answer:

Given that,

The 5pound bag of organic brown rice = $12.99.

For each bag = $12.99/5 = $2.598.

The 12pound bag of organic brown rice = $31.99

For each bag = $31.99/12 = $2.665

Therefore a 5pound bag of organic rice is better to buy because the cost of a 5pound bag is less than the cost of a 12pound bag.