Pre-Algebra Alıştırma Soruları — ACT — Sayfa 21

Soru 401Soru →

A school library receives a shipment of $12$ new books, consisting of $5$ science fiction novels, $4$ historical fiction novels, and $3$ biography books. A librarian wants to choose $3$ books to feature on a display shelf. How many different combinations of $3$ books can be formed that contain exactly $2$ science fiction novels and $1$ biography book?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The total number of combinations containing exactly 2 science fiction novels and 1 biography book is 30.

To find the total number of combinations, first determine the number of ways to choose

2

science fiction novels from

5

available:

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

. Next, determine the number of ways to choose

1

biography book from

3

available:

\binom{3}{1} = 3

. Finally, apply the Fundamental Counting Principle by multiplying the counts together:

10 \times 3 = 30

.

Adım Adım Çözüm

1

Calculate the combinations for selecting the science fiction novels.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

Select

2

novels out of the

5

available science fiction novels without regard to order.

2

Calculate the combinations for selecting the biography book.

\binom{3}{1} = 3

Select

1

book out of the

3

available biography books.

3

Apply the Fundamental Counting Principle to find total combinations.

10 \times 3 = 30

Multiply the combinations calculated for each independent selection.

Anahtar Kavram

Combinations and the Fundamental Counting Principle

Soru 402Soru →

A project team of $3$ software developers is to be randomly selected from a pool of $4$ frontend developers and $6$ backend developers. What is the probability that the selected team will consist of exactly $1$ frontend developer and $2$ backend developers?

\frac{1}{6}

\frac{3}{10}

\frac{19}{120}

\frac{1}{2}

\frac{3}{25}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{2}$

Cevap

The probability that the team consists of exactly 1 frontend developer and 2 backend developers is

\frac{1}{2}

.

To find the probability, divide the number of ways to select 1 frontend developer and 2 backend developers by the total number of ways to choose any 3 developers from 10. There are

\binom{4}{1} = 4

ways to select 1 frontend developer and

\binom{6}{2} = 15

ways to select 2 backend developers. Multiplying these yields

4 \times 15 = 60

favorable team combinations. The total number of ways to choose 3 developers from 10 is

\binom{10}{3} = 120

. Thus, the probability is

\frac{60}{120} = \frac{1}{2}

.

Adım Adım Çözüm

1

Calculate the total number of possible 3-developer teams from the pool of 10 developers.

\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

total outcomes.

The total number of ways to choose 3 people out of 10 without regard to order is given by combinations.

2

Calculate the number of ways to select 1 frontend developer from 4 and 2 backend developers from 6.

\binom{4}{1} \times \binom{6}{2} = 4 \times \frac{6 \times 5}{2 \times 1} = 4 \times 15 = 60

favorable outcomes.

By the Fundamental Counting Principle, multiply the number of ways to choose the frontend developer by the number of ways to choose the backend developers.

3

Divide the number of favorable outcomes by the total possible outcomes to find the probability.

\text{Probability} = \frac{60}{120} = \frac{1}{2}

.

Probability is defined as the ratio of favorable outcomes to total outcomes in a uniform sample space.

Anahtar Kavram

Probability of Compound Events using Combinations

Soru 403Soru →

A bakery allows customers to customize a fruit tart by selecting $1$ crust type from $3$ available choices, $1$ filling type from $4$ available choices, and $2$ different fruit toppings from $5$ available choices. How many different fruit tart combinations are possible?

17

60

120

240

300

Cevabı ve açıklamayı göster

Cevap: $120$

Cevap

The total number of different fruit tart combinations possible is 120.

To find the total number of unique fruit tart combinations, apply the Fundamental Counting Principle across all decision stages. Selecting 1 crust out of 3 yields 3 options. Selecting 1 filling out of 4 yields 4 options. Selecting 2 different fruit toppings out of 5 requires choosing combinations without regard to order, which is calculated as 5 choose 2 = 10. Multiplying the options from each stage gives 3 * 4 * 10 = 120 total combinations.

Adım Adım Çözüm

1

Determine the number of ways to choose the crust and filling.

There are

3

crust choices and

4

filling choices.

Each selection stage is independent.

2

Calculate the number of ways to choose

2

different fruit toppings from

5

available options.

\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

choices.

The order in which the two toppings are selected does not matter, so combinations must be used.

3

Apply the Fundamental Counting Principle to find the total combinations.

3 \times 4 \times 10 = 120

.

Multiply the number of choices for each independent step.

Anahtar Kavram

Fundamental Counting Principle and Combinations

Tahmini Süre:1m 0s

Soru 404Soru →

An art gallery manager is arranging 6 unique paintings side by side in a single line on a display wall. Two specific paintings created by the same artist must be placed directly next to each other. How many different line arrangements of the 6 paintings are possible?

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

The total number of different line arrangements is 240.

To satisfy the constraint that two specific paintings must be adjacent, treat them as a single combined block. This reduces the problem to arranging 5 items (the single block plus the other 4 paintings), which can be ordered in 5! = 120 ways. Within the block, the two paintings can be ordered in 2! = 2 ways. Applying the Fundamental Counting Principle gives 120 * 2 = 240 distinct arrangements.

Adım Adım Çözüm

1

Group the two specific paintings together as a single block.

There are 5 units to arrange (1 block of 2 paintings plus 4 individual paintings).

Since the two paintings must remain adjacent, treating them as a single item guarantees they will stay together.

2

Calculate the permutations of the 5 units.

5! = 5 * 4 * 3 * 2 * 1 = 120 ways.

There are 5 distinct units being arranged in a straight line.

3

Determine the internal arrangements of the grouped pair.

2! = 2 * 1 = 2 ways.

The two specific paintings within the block can switch positions (Painting A then B, or Painting B then A).

4

Multiply the arrangements together.

120 * 2 = 240 total arrangements.

By the Fundamental Counting Principle, total arrangements equal the number of ways to place the units multiplied by the internal arrangements of the restricted pair.

Anahtar Kavram

Permutations with Restrictions (Grouped Elements)

Tahmini Süre:1m 15s

Soru 405Soru →

A box contains $30$ raffle tickets numbered $1$ through $30$ . If one ticket is selected at random from the box, what is the probability that the number on the selected ticket is a multiple of $4$ or a multiple of $6$ ?

\frac{1}{15}

\frac{1}{5}

\frac{1}{3}

\frac{2}{5}

\frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{3}$

Cevap

The probability that the number on the selected ticket is a multiple of

4

or a multiple of

6

is

\frac{1}{3}

.

The correct answer is determined by applying the addition rule for non-mutually exclusive events:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

. Out of

30

tickets,

7

are multiples of

4

and

5

are multiples of

6

. The numbers

12

and

24

are multiples of both

4

and

6

, meaning they belong to the intersection. Calculating the total number of unique favorable tickets gives

7 + 5 - 2 = 10

. Dividing

10

favorable outcomes by the

30

total tickets yields a probability of

\frac{10}{30} = \frac{1}{3}

.

Adım Adım Çözüm

1

Identify the total number of possible outcomes.

There are

30

total tickets in the box, so the sample space size is

30

.

Each ticket numbered 1 through 30 is equally likely to be selected.

2

Count the outcomes for each individual event.

Multiples of

4

:

\{4, 8, 12, 16, 20, 24, 28\} \rightarrow 7

outcomes.
Multiples of

6

:

\{6, 12, 18, 24, 30\} \rightarrow 5

outcomes.

Listing or dividing

30

by

4

and

6

determines how many tickets satisfy each separate condition.

3

Identify and count the overlapping outcomes (multiples of both

4

and

6

).

Multiples of both

4

and

6

are multiples of

\text{LCM}(4, 6) = 12

, which are

\{12, 24\} \rightarrow 2

outcomes.

These numbers are contained in both individual lists and will be double-counted if not accounted for.

4

Apply the Principle of Inclusion-Exclusion to find the total number of favorable outcomes.

\text{Favorable outcomes} = 7 + 5 - 2 = 10.

Subtracting the intersection prevents double-counting the tickets that satisfy both conditions.

5

Calculate the probability.

P(\text{multiple of } 4 \text{ or } 6) = \frac{10}{30} = \frac{1}{3}.

Probability is the ratio of favorable outcomes to total possible outcomes.

Anahtar Kavram

Probability of Non-Mutually Exclusive Events (Inclusion-Exclusion Principle)

Tahmini Süre:1m 0s

Soru 406Soru →

A student is creating a flag design consisting of 3 vertical stripes in a row. The student can choose from 5 different colors: Red, Blue, Green, Yellow, and White. To ensure visual contrast, any 2 adjacent stripes must be different colors. How many different flag designs can be created under these rules?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80

To find the total number of flag designs, apply the Fundamental Counting Principle across the three sequential stripe choices. The first stripe can be any of the 5 colors. The second stripe cannot match the first stripe's color, leaving 4 options. The third stripe cannot match the second stripe's color, which also leaves 4 options (since it can re-use the first stripe's color). Multiplying these choices together gives

5 \times 4 \times 4 = 80

unique flag designs.

Adım Adım Çözüm

1

Determine choices for the first stripe

5 choices

Any of the 5 available colors can be selected for the first stripe.

2

Determine choices for the second stripe

4 choices

The second stripe must differ in color from the first stripe.

3

Determine choices for the third stripe

4 choices

The third stripe must differ in color from the second stripe (it may be the same color as the first stripe).

4

Apply the Fundamental Counting Principle

5 \times 4 \times 4 = 80

Multiply the options available at each stage to calculate the total number of distinct flag designs.

Anahtar Kavram

Fundamental Counting Principle with adjacent restrictions

Tahmini Süre:1m 15s

Soru 407Soru →

A class of $40$ students has a ratio of boys to girls of $3:5$ . How many more girls than boys are in the class?

5

8

10

15

25

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

There are 10 more girls than boys in the class.

To find the difference between the number of girls and boys, we first determine the value of one part of the ratio. Since the ratio of boys to girls is

3:5

, there are

3 + 5 = 8

total parts. With

40

students in total, each part corresponds to

40 \div 8 = 5

students. The difference between the ratio parts is

5 - 3 = 2

parts. Multiplying this difference by the value of one part gives

2 \times 5 = 10

more girls than boys.

Adım Adım Çözüm

1

Find the total number of parts in the ratio by adding the parts for boys and girls.

3 + 5 = 8

parts

To determine how the total number of students is divided, we need the sum of the ratio parts.

2

Calculate how many students represent one part of the ratio by dividing the total number of students by the total parts.

40 \div 8 = 5

students per part

This establishes the scale factor for the ratio to convert parts into actual student counts.

3

Find the difference in ratio parts between girls and boys.

5 - 3 = 2

parts

The question asks for the difference in the number of girls and boys, which corresponds to the difference in their ratio parts.

4

Multiply the difference in parts by the number of students per part.

2 \times 5 = 10

students

This gives the final difference in the number of girls and boys.

Anahtar Kavram

Solving part-to-part ratio problems using total quantities

Alternatif Yöntem

Alternatively, calculate the actual number of boys and girls first. The number of boys is

\frac{3}{8} \times 40 = 15

and the number of girls is

\frac{5}{8} \times 40 = 25

. Subtracting the number of boys from the number of girls yields

25 - 15 = 10

.

Tahmini Süre:45s

Soru 408Soru →

If the least common multiple of a positive integer $n$ and 15 is 90, how many possible values are there for $n$ ?

1

2

3

4

6

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

The correct answer is 2. The prime factorization of 15 is

3^1 \times 5^1

, and the prime factorization of 90 is

2^1 \times 3^2 \times 5^1

. The least common multiple (LCM) of two numbers is found by taking the highest power of each prime factor present in their factorizations. Since 90 contains

2^1

, and 15 does not,

n

must contain exactly

2^1

. Since 90 contains

3^2

, and 15 only contains

3^1

,

n

must contain exactly

3^2

. Since both 90 and 15 contain

5^1

, the power of 5 in

n

can be either 0 or 1. This results in two possible values:

2^1 \times 3^2 \times 5^0 = 18

and

2^1 \times 3^2 \times 5^1 = 90

.

Adım Adım Çözüm

1

Write out the prime factorizations of 15 and 90.

15 = 3^1 \times 5^1

and

90 = 2^1 \times 3^2 \times 5^1

.

To analyze the relationship between

n

, 15, and their least common multiple (LCM), we must look at their prime factor components.

2

Determine the constraints on the exponents of the prime factors of

n

using the definition of LCM.

The prime factorization of

n

must be of the form

2^a \times 3^b \times 5^c

. For the LCM of

n

and 15 to be 90, the exponent of each prime factor in the LCM is the maximum of its exponents in

n

and 15. Thus, we must have

a = 1

,

b = 2

, and

c \in \{0, 1\}

.

The LCM of two numbers takes the highest power of each prime factor present in both numbers.

3

Count the number of possible values for

n

.

Since

a

has 1 choice,

b

has 1 choice, and

c

has 2 choices, there are

1 \times 1 \times 2 = 2

possible values for

n

(

n = 2^1 \times 3^2 \times 5^0 = 18

and

n = 2^1 \times 3^2 \times 5^1 = 90

).

This gives the total number of distinct integers satisfying the condition.

Anahtar Kavram

The least common multiple of two numbers is found by taking the highest power of each prime factor present in the prime factorizations of both numbers.

Alternatif Yöntem

Alternatively, you can test the positive factors of 90 that are not factors of 15. The positive factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. By computing the least common multiple of each factor with 15, we find that only LCM(18, 15) = 90 and LCM(90, 15) = 90 satisfy the condition.

Tahmini Süre:1m 0s

Soru 409Soru →

A scientist records the daily growth of a plant in millimeters over 5 days in the table below:

Day	Growth (mm)
Monday	6
Tuesday	12
Wednesday	4
Thursday	11
Friday	7

What is the median daily growth, in millimeters, of the plant over these 5 days?

4

6

7

8

10

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

To find the median daily growth, first list the growth values in order from least to greatest: 4, 6, 7, 11, 12. Since there are 5 values, the median is the middle value (the third number), which is 7.

Adım Adım Çözüm

1

List the daily growth values from the table.

6, 12, 4, 11, 7

Identify the raw data points that need to be analyzed.

2

Arrange the growth values in order from least to greatest.

4, 6, 7, 11, 12

The median is the middle value of an ordered dataset.

3

Find the middle value in the sorted list of 5 numbers.

7

Since there are 5 numbers, the median is the 3rd number in the ordered list, which is 7.

Anahtar Kavram

The median of a set of data is the middle number when the data values are arranged in order from least to greatest.

Tahmini Süre:45s

Soru 410Soru →

An online streaming service charges a flat monthly subscription fee of $\$12$ plus $\$2.50$ for each premium movie rented. The total monthly cost, in dollars, for a user who rents $m$ premium movies is represented by the expression $12 + 2.50m$ . What is the total monthly cost, in dollars, for a user who rents $6$ premium movies?

\$15.00

\$20.50

\$27.00

\$74.50

\$87.00

Cevabı ve açıklamayı göster

Cevap: $\$27.00$

Cevap

\$27.00

The correct answer is

\$27.00

. To find the total monthly cost when a user rents

6

premium movies, we substitute

m = 6

into the given expression

12 + 2.50m

. Applying the order of operations, we first perform the multiplication:

2.50 \times 6 = 15

. Next, we add the flat monthly fee:

12 + 15 = 27

. Therefore, the total monthly cost is

\$27.00

.

Adım Adım Çözüm

1

Identify the given algebraic expression for the monthly cost and the value to substitute.

Expression:

12 + 2.50m

; value of

m

:

6

The variable

m

represents the number of premium movies rented, which is given as

6

in the question.

2

Substitute

m = 6

into the expression.

Expression becomes

12 + 2.50(6)

Substitution allows us to evaluate the expression for the specific scenario described.

3

Apply the order of operations by performing multiplication before addition.

2.50 \times 6 = 15

Multiplication must be completed before addition according to arithmetic rules.

4

Add the flat fee to the total movie rental cost.

12 + 15 = 27

This completes the evaluation of the expression, yielding the total monthly cost of

\$27.00

.

Anahtar Kavram

Evaluating algebraic expressions

Tahmini Süre:45s

Soru 411Soru →

An online bookstore sells paperback novels for $12$ dollars each. The total cost, in dollars, to purchase $n$ novels and have them shipped to a customer's home is given by the expression $12n + 8$ . What is the total cost, in dollars, to purchase $4$ novels and have them shipped?

24

40

56

80

144

Cevabı ve açıklamayı göster

Cevap: 56

Cevap

The total cost to purchase 4 novels and have them shipped is 56 dollars.

The correct answer is 56. Substituting the number of novels (

n = 4

) into the cost expression

12n + 8

gives

12(4) + 8

. Following the order of operations, we first multiply 12 by 4 to get 48, and then add 8, resulting in 56.

Adım Adım Çözüm

1

Substitute the number of novels,

n = 4

, into the given cost expression,

12n + 8

.

The expression becomes

12(4) + 8

.

To find the total cost for a specific number of novels, we replace the variable representing the number of novels with the given value.

2

Apply the order of operations (PEMDAS/BODMAS) to evaluate the expression by multiplying before adding.

12 \times 4 = 48

, then

48 + 8 = 56

.

Multiplication must be performed before addition according to standard mathematical rules.

Anahtar Kavram

Evaluating algebraic expressions using the order of operations

Soru 412Soru →

A technology company has three departments: Sales, Marketing, and Engineering. The table below shows the number of employees in the Sales and Marketing departments, along with the mean annual salary for each of the three departments.

Department	Number of Employees	Mean Annual Salary
Sales	$8$	$\$45,000$
Marketing	$12$	$\$55,000$
Engineering	$N$	$\$85,000$

If the mean annual salary of all employees across all three departments is $\$68,000$ , what is the total number of employees in the company?

20

24

30

40

45

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40

To find the total number of employees, we first calculate the total salary for each department. The Sales department total is

8 \times \$45,000 = \$360,000

. The Marketing department total is

12 \times \$55,000 = \$660,000

. If there are

N

employees in Engineering, their total salary is

N \times \$85,000 = \$85,000N

. The total salary for the entire company is

\$360,000 + \$660,000 + \$85,000N = \$1,020,000 + \$85,000N

. The total number of employees is

8 + 12 + N = 20 + N

. Since the overall mean salary is

\$68,000

, we set up the weighted average equation:

\frac{1,020,000 + 85,000N}{20 + N} = 68,000

. Solving for

N

gives

1,020,000 + 85,000N = 68,000(20 + N)

, which simplifies to

1,020,000 + 85,000N = 1,360,000 + 68,000N

. Subtracting

68,000N

and

1,020,000

from both sides yields

17,000N = 340,000

, so

N = 20

. The total number of employees in the company is the sum of the employees in all three departments:

8 + 12 + 20 = 40

.

Adım Adım Çözüm

1

Calculate the total combined annual salary for the Sales and Marketing departments.

Sales total:

8 \times \$45,000 = \$360,000

. Marketing total:

12 \times \$55,000 = \$660,000

. Combined total: \

360,000 + \$660,000 = \$1,020,000

.

To find the overall mean, we need the sum of all salaries in the company.

2

Express the total company salary and total number of employees in terms of

N

.

Total salary:

\$1,020,000 + 85,000N

. Total employees:

8 + 12 + N = 20 + N

.

These terms will be used to set up the weighted average formula.

3

Set up the weighted average equation and solve for

N

.

Equation:

\frac{1,020,000 + 85,000N}{20 + N} = 68,000

. Multiplying both sides by

20 + N

gives

1,020,000 + 85,000N = 1,360,000 + 68,000N

. Subtracting

68,000N

and

1,020,000

from both sides yields

17,000N = 340,000

, which simplifies to

N = 20

.

Solving this equation gives the number of employees in the Engineering department.

4

Calculate the total number of employees in the company.

Total employees =

8 + 12 + 20 = 40

.

The question asks for the total number of employees, which is the sum of employees in all three departments.

Anahtar Kavram

Weighted Average (Mean) with Variables

Tahmini Süre:2m 0s

Soru 413Soru →

A pharmaceutical laboratory formulates a liquid medication by combining Active Ingredient X, Stabilizer Y, and Purified Water in a ratio of $3 : 4 : 8$ by volume. A technician needs to prepare a batch of this medication that contains exactly $360\text{ mL}$ of Stabilizer Y. How many total milliliters of liquid medication will this batch contain?

675\text{ mL}

720\text{ mL}

990\text{ mL}

1,350\text{ mL}

5,400\text{ mL}

Cevabı ve açıklamayı göster

Cevap: $1,350\text{ mL}$

Cevap

1,350\text{ mL}

To find the total batch volume, determine the value of one part in the ratio. The three components (Active Ingredient X, Stabilizer Y, and Purified Water) are mixed in a ratio of

3 : 4 : 8

, giving a total of

3 + 4 + 8 = 15

parts. Stabilizer Y represents

4

of these parts and has a volume of

360\text{ mL}

. Dividing

360

by

4

gives

90\text{ mL}

per part. Multiplying

90\text{ mL}

by the total

15

parts yields

1,350\text{ mL}

.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

Total parts =

3 + 4 + 8 = 15

parts.

The complete mixture is made up of the sum of the ratio components.

2

Determine the volume represented by a single ratio part.

Value of

1

part =

\frac{360\text{ mL}}{4} = 90\text{ mL}

.

Stabilizer Y accounts for

4

parts of the total ratio and has a volume of

360\text{ mL}

.

3

Calculate the total volume of the batch.

Total volume =

15 \times 90\text{ mL} = 1,350\text{ mL}

.

Multiply the single-part volume by the total number of parts.

Anahtar Kavram

Solving Part-to-Total Proportions with Multiple Components

Tahmini Süre:1m 15s

Soru 414Soru →

A quality control laboratory tested four battery models to measure the proportion of initial charge retained after 24 hours of continuous operation. The recorded results for each model are listed below:

- Model Alpha: $\frac{13}{20}$
- Model Beta: $64.5\%$
- Model Gamma: $\frac{5}{8}$
- Model Delta: $0.638$

Which of the following represents the battery models arranged in order from least to greatest proportion of remaining charge?

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from least to greatest proportion of remaining charge is Model Gamma (

\frac{5}{8}

), Model Delta (

0.638

), Model Beta (

64.5\%

), and Model Alpha (

\frac{13}{20}

).

Converting each model's retained charge to decimal form gives Model Gamma

= \frac{5}{8} = 0.625

, Model Delta

= 0.638

, Model Beta

= 64.5\% = 0.645

, and Model Alpha

= \frac{13}{20} = 0.650

. Comparing these values yields

0.625 < 0.638 < 0.645 < 0.650

, which matches the order Model Gamma, Model Delta, Model Beta, Model Alpha.

Adım Adım Çözüm

1

Convert each given fraction, percentage, and decimal into equivalent decimal values for direct comparison.

Model Alpha

= \frac{13}{20} = 0.650

; Model Beta

= 64.5\% = 0.645

; Model Gamma

= \frac{5}{8} = 0.625

; Model Delta

= 0.638

.

Converting all values to decimals standardizes the quantities so they can be compared easily by place value.

2

Compare the resulting decimal numbers from least to greatest.

0.625 < 0.638 < 0.645 < 0.650

Comparing digits from left to right: all have

0.6

in the tenths place; comparing the hundredths place shows

2 < 3 < 4 < 5

.

3

Map the sorted decimal values back to their original model designations.

Model Gamma (

\frac{5}{8}

), Model Delta (

0.638

), Model Beta (

64.5\%

), Model Alpha (

\frac{13}{20}

).

This establishes the required sequence from least to greatest.

Anahtar Kavram

Converting fractions, decimals, and percentages to a common format (decimals) to order rational numbers.

Soru 415Soru →

A landscape architect is designing a public park with a total area of $14,400\text{ square feet}$ . The plan allocates space for a lawn, flowerbeds, and paved walkways in the ratio $7 : 3 : 2$ , respectively. How many more square feet of area are dedicated to the lawn than to the flowerbeds?

3,600

4,800

5,760

6,000

8,400

Cevabı ve açıklamayı göster

Cevap: 4,800

Cevap

4,800 square feet

The sum of all ratio parts is

7 + 3 + 2 = 12

. Dividing the total park area of

14,400\text{ sq ft}

by

12

gives

1,200\text{ sq ft}

per ratio part. The lawn accounts for

7

parts (

8,400\text{ sq ft}

) and the flowerbeds account for

3

parts (

3,600\text{ sq ft}

). Subtracting the flowerbed area from the lawn area gives

8,400 - 3,600 = 4,800\text{ sq ft}

. Alternatively, taking the direct difference of

7 - 3 = 4

parts and multiplying by

1,200\text{ sq ft}

gives

4,800\text{ sq ft}

.

Adım Adım Çözüm

1

Find the total number of equal ratio parts.

7 + 3 + 2 = 12\text{ parts}

The complete park area is divided into portions defined by the sum of all terms in the ratio.

2

Calculate the area corresponding to one ratio part.

\frac{14,400}{12} = 1,200\text{ square feet per part}

Dividing the total area by the total number of parts determines the size of a single unit part.

3

Determine the difference in ratio parts between the lawn and the flowerbeds.

7 - 3 = 4\text{ parts}

The lawn receives 7 parts and the flowerbeds receive 3 parts.

4

Multiply the difference in parts by the area per part.

4 \times 1,200 = 4,800\text{ square feet}

Multiplying the difference in ratio units by the unit area yields the total excess area of the lawn over the flowerbeds.

Anahtar Kavram

Solving multi-part ratio word problems using unit rate allocation per ratio part.

Tahmini Süre:1m 15s

Soru 416Soru →

A woodworker starts with a wooden board of length $L$ inches. After cutting off a piece measuring $16.5$ inches to build a shelf bracket, the remaining portion of the board is $48.5$ inches long. Which of the following equations can be used to find $L$ , the original length of the wooden board in inches?

L - 16.5 = 48.5

L + 16.5 = 48.5

16.5L = 48.5

\frac{L}{16.5} = 48.5

48.5 - L = 16.5

Cevabı ve açıklamayı göster

Cevap: $L - 16.5 = 48.5$

Cevap

L - 16.5 = 48.5

The original length of the board is

L

inches. Cutting off a piece measuring

16.5

inches decreases the total length by

16.5

, which is written algebraically as

L - 16.5

. Since the remaining length is given as

48.5

inches, setting these equal produces the equation

L - 16.5 = 48.5

.

Adım Adım Çözüm

1

Identify the given variables and quantities in the scenario

Initial length =

L

, length cut off =

16.5

, remaining length =

48.5

Establishing clear variables allows systematic setup of the algebraic model.

2

Translate the physical action into a mathematical operation

Cutting off a piece represents subtraction from the starting quantity:

L - 16.5

Removal of a part from a whole corresponds to subtraction.

3

Equate the expression for remaining length to the given remaining value

L - 16.5 = 48.5

The remaining length after cutting is stated to be

48.5

inches.

Anahtar Kavram

Translating verbal removal scenarios into one-step subtraction equations

Tahmini Süre:1m 0s

Soru 417Soru →

A venue manager allocates a total hospitality budget of $d$ dollars for an upcoming music festival. If the manager divides this budget equally among 14 performing artists, each artist receives exactly $\$185$ . Which of the following equations correctly models this situation to find the total budget $d$ ?

\frac{d}{14} = 185

14d = 185

d - 14 = 185

d + 185 = 14

\frac{185}{d} = 14

Cevabı ve açıklamayı göster

Cevap: $\frac{d}{14} = 185$

Cevap

The correct equation is

\frac{d}{14} = 185

, which expresses that the total budget

d

divided equally among 14 artists yields

\$185

per artist.

Equal distribution of a total quantity

d

across 14 recipients is mathematically represented by dividing

d

by 14. Setting this quotient equal to the per-person amount of

\$185

yields the equation

\frac{d}{14} = 185

.

Adım Adım Çözüm

1

Identify the given quantities and variable

Total budget =

d

, number of equal shares =

14

, amount per share =

\$185

Setting up an equation requires matching the verbal scenario to mathematical operations.

2

Translate equal sharing into an algebraic operation

The total budget

d

divided by

14

is written as

\frac{d}{14}

Dividing a total quantity into equal parts corresponds to division.

3

Equate the expression to the given share amount

\frac{d}{14} = 185

Each share is given as

\$185

.

Anahtar Kavram

Translating verbal division scenarios into one-step linear equations

Soru 418Soru →

At an auto dealership, $\frac{2}{5}$ of the vehicles in the inventory are sedans, $35\%$ are SUVs, and the remaining 45 vehicles are trucks. What is the total number of vehicles in the dealership's inventory?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

The total number of vehicles in the dealership's inventory is 180.

Converting

\frac{2}{5}

to a percentage yields

40\%

. Adding the

40\%

sedans and

35\%

SUVs shows that

75\%

of the inventory consists of sedans and SUVs. Consequently, the remaining

25\%

of the inventory consists of trucks. Since there are 45 trucks,

25\%

of the total inventory equals 45. Dividing 45 by

0.25

yields the total inventory of 180 vehicles.

Adım Adım Çözüm

1

Convert fraction of sedans to a percentage

Sedans represent

40\%

of the total inventory.

To combine the amounts easily, convert

\frac{2}{5}

to percentage form:

\frac{2}{5} = 0.40 = 40\%

.

2

Calculate the combined percentage of sedans and SUVs

Sedans and SUVs together make up

75\%

of the inventory.

Add the two percentages:

40\% + 35\% = 75\%

.

3

Determine the remaining percentage representing trucks

Trucks make up

25\%

of the inventory.

The entire inventory equals

100\%

, so

100\% - 75\% = 25\%

.

4

Solve for the total number of vehicles

The total inventory is 180 vehicles.

Set up the equation

25\% \text{ of Total} = 45

, which gives

0.25 \times \text{Total} = 45

, so

\text{Total} = \frac{45}{0.25} = 180

.

Anahtar Kavram

Converting fractions to percentages to find an unknown total quantity from a remaining part

Soru 419Soru →

At a community theater, ticket sales for a weekend musical were divided among three ticket categories: advance online purchases, box office walk-ups, and discounted student passes. Advance online purchases accounted for $\frac{2}{5}$ of total ticket sales, and box office walk-up sales accounted for $35\%$ of total ticket sales. If the remaining 125 tickets sold were discounted student passes, what was the total number of tickets sold for the musical?

375

500

167

313

94

Cevabı ve açıklamayı göster

Cevap: 500

Cevap

The total number of tickets sold for the musical was 500.

Advance online tickets represent

\frac{2}{5} = 40\%

of the total. Adding the

35\%

from box office walk-ups gives

75\%

of total sales accounted for. The remaining

25\%

of sales corresponds to the 125 student passes. Setting

25\%

of the total equal to 125 and solving gives a total of 500 tickets.

Adım Adım Çözüm

1

Convert the fraction of advance online tickets to a percentage.

\frac{2}{5} = \frac{40}{100} = 40\%

Converting all proportions to percentages allows for direct comparison and addition.

2

Calculate the combined percentage of advance online and box office ticket sales.

40\% + 35\% = 75\%

Adding the two known categories determines the total portion of non-student ticket sales.

3

Find the percentage corresponding to the discounted student passes.

100\% - 75\% = 25\%

The sum of all categories must equal

100\%

of total sales.

4

Set up an equation relating the remaining percentage to the total ticket count

T

.

0.25 \times T = 125 \implies T = \frac{125}{0.25} = 500

Dividing the part by its corresponding percentage rate yields the whole amount.

Anahtar Kavram

Solving multi-step word problems involving conversions between fractions, percentages, and whole numbers.

Tahmini Süre:1m 30s

Pre-Algebra

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