Translating and Solving Algebraic Word Problems Alıştırma Soruları — ACT — Sayfa 3

Soru 41Soru →

A food truck selling gourmet grilled cheese sandwiches has a daily fixed operating cost of $120$ . Each sandwich costs $2.50$ to make and is sold for $6.50$ . What is the minimum number of sandwiches the food truck must sell in one day to make a net profit of at least $180$ ?

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Cevap: 75

Cevap

The food truck must sell a minimum of 75 sandwiches to make a net profit of at least $180.

Representing the number of sandwiches sold as

x

, the total revenue is

6.50x

and the total cost is

120 + 2.50x

. The net profit is the difference between revenue and cost:

6.50x - (120 + 2.50x)

, which simplifies to

4x - 120

. Setting up the inequality for a profit of at least

180

gives

4x - 120 \geq 180

. Solving for

x

gives

4x \geq 300

, which simplifies to

x \geq 75

. Therefore, the minimum number of sandwiches that must be sold is

75

.

Adım Adım Çözüm

1

Define the variable and write expressions for revenue and cost.

Let

x

represent the number of sandwiches sold. Total Revenue =

6.50x

and Total Cost =

120 + 2.50x

.

Defining the variable and translating the verbal descriptions of revenue and cost into algebraic expressions is necessary to model the profit.

2

Formulate the net profit expression.

Net Profit = Total Revenue - Total Cost =

6.50x - (120 + 2.50x) = 4x - 120

.

Net profit is calculated by subtracting all fixed and variable costs from the total revenue.

3

Set up and solve the linear inequality.

4x - 120 \geq 180 \Rightarrow 4x \geq 300 \Rightarrow x \geq 75

.

To find the minimum number of sandwiches needed to reach a target profit of at least

180

, we solve the inequality

4x - 120 \geq 180

for

x

.

Anahtar Kavram

Translating a real-world scenario into a linear inequality and solving for the unknown variable.

Soru 42Soru →

The ratio of the number of students in a school's science club to the number of students in its art club is $3:5$ . There are $8$ more students in the art club than in the science club. If $x$ students from the art club leave to join the science club, the ratio of the number of science club members to the number of art club members becomes $3:1$ . What is the value of $x$ ?

4

6

12

24

48

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Cevap: 12

Cevap

12

The correct answer is 12. Let the initial number of students in the science and art clubs be

3k

and

5k

, respectively. Since there are 8 more students in the art club, we set up the equation

5k = 3k + 8

, which simplifies to

2k = 8

, giving

k = 4

. This means there are initially 12 science club members and 20 art club members. When

x

students transfer from the art club to the science club, the new sizes are

12 + x

and

20 - x

. Setting up the new ratio gives

\frac{12 + x}{20 - x} = 3

. Solving this equation yields

12 + x = 60 - 3x

, which simplifies to

4x = 48

, resulting in

x = 12

.

Adım Adım Çözüm

1

Represent the initial number of students in each club using the given ratio.

Let the number of science club members be

3k

and the number of art club members be

5k

, where

k

is a positive constant.

The ratio of science club members to art club members is

3:5

.

2

Set up and solve a linear equation to find the value of

k

and the starting number of members in each club.

5k = 3k + 8 \Rightarrow 2k = 8 \Rightarrow k = 4

. Thus, the science club has

3(4) = 12

members and the art club has

5(4) = 20

members.

There are

8

more students in the art club than in the science club, so the difference between the two groups is

8

.

3

Write the new member counts after

x

students transfer and set up the ratio equation.

The new science club size is

12 + x

and the new art club size is

20 - x

. The new ratio equation is

\frac{12 + x}{20 - x} = \frac{3}{1}

.

The

x

students leave the art club (subtracting

x

) and join the science club (adding

x

), resulting in a new ratio of

3:1

.

4

Solve the rational equation for

x

.

12 + x = 3(20 - x) \Rightarrow 12 + x = 60 - 3x \Rightarrow 4x = 48 \Rightarrow x = 12

.

Cross-multiplying and isolating

x

yields the number of students who transferred.

Anahtar Kavram

Translating verbal statements involving ratios and changes in quantities into solvable linear algebraic equations.

Tahmini Süre:1m 30s

Soru 43Soru →

A commercial building has two water reservoirs. Reservoir $X$ contains $1,200$ gallons of water and is draining at a constant rate of $18$ gallons per minute. Reservoir $Y$ contains $360$ gallons of water and is being filled at a constant rate of $22$ gallons per minute. After how many minutes will both reservoirs contain the exact same amount of water?

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Cevap: 21

Cevap

The two reservoirs will contain the same amount of water after 21 minutes.

The correct answer is 21 minutes. By setting the expressions for the volume of both reservoirs equal (

1,200 - 18t = 360 + 22t

) and isolating the variable, we find

40t = 840

, which simplifies to

t = 21

.

Adım Adım Çözüm

1

Translate the physical scenario for Reservoir

X

into an algebraic expression.

1,200 - 18t

Reservoir

X

starts with

1,200

gallons and loses

18

gallons per minute over

t

minutes.

2

Translate the physical scenario for Reservoir

Y

into an algebraic expression.

360 + 22t

Reservoir

Y

starts with

360

gallons and gains

22

gallons per minute over

t

minutes.

3

Set the two expressions equal to each other and solve for

t

.

1,200 - 18t = 360 + 22t \implies 840 = 40t \implies t = 21

Equating the two volume expressions allows us to find the time

t

at which the volumes are equal.

Anahtar Kavram

Translating and Solving Algebraic Word Problems

Alternatif Yöntem

Instead of solving algebraically, one can check the rates of change relative to each other. The distance between the initial volumes is

1,200 - 360 = 840

gallons. Since they are moving toward each other (one draining, one filling), their relative rate of convergence is

18 + 22 = 40

gallons per minute. Dividing the total volume difference by the rate of convergence gives

840 / 40 = 21

minutes.

Tahmini Süre:1m 30s

Soru 44Soru →

A shipping company charges a flat fee of $15$ dollars plus $2.50$ dollars per pound for the first $10$ pounds of a package's weight. For any weight exceeding $10$ pounds, the rate is $4.00$ dollars per pound. If the total shipping charge for a package was $59.00$ dollars, what was the total weight of the package, in pounds?

11.0

11.6

14.75

17.6

18.5

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Cevap: $14.75$

Cevap

The package's total weight was 14.75 pounds.

The correct answer of 14.75 pounds is found by setting up the equation representing the total shipping cost. First, calculate the cost for the first 10 pounds, which is the flat fee of 15.00 dollars plus 2.50 dollars per pound for 10 pounds: 15 + 2.50(10) = 40.00 dollars. Since the total charge of 59.00 dollars is greater than 40.00 dollars, the package must weigh more than 10 pounds. Let w be the total weight of the package. The remaining weight exceeding 10 pounds is w - 10, which is charged at 4.00 dollars per pound. Setting up the equation: 40 + 4(w - 10) = 59. Solving for w gives: 4(w - 10) = 19, which simplifies to w - 10 = 4.75, so w = 14.75.

Adım Adım Çözüm

1

Calculate the cost for a package weighing exactly 10 pounds.

The cost is 15 + 2.50 * 10 = 40.00 dollars.

To determine whether the package exceeds 10 pounds by comparing it to the total charge of 59.00 dollars.

2

Set up an equation for the total cost where the weight w exceeds 10 pounds.

40 + 4.00 * (w - 10) = 59.00

To express the cost of the first 10 pounds plus the cost of the excess weight at 4.00 dollars per pound.

3

Solve the equation for the total weight w.

4 * (w - 10) = 19 -> w - 10 = 4.75 -> w = 14.75

To isolate and find the value of the variable representing the package weight.

Anahtar Kavram

Translating and Solving Algebraic Word Problems

Tahmini Süre:1m 30s

Translating and Solving Algebraic Word Problems

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