Pre-Algebra Alıştırma Soruları — ACT — Sayfa 18

Soru 341Soru →

A specialty coffee roasting facility packages a batch of $c$ pounds of dark roast coffee beans into retail bags. Each bag is filled with exactly $1.5$ pounds of coffee. If the facility completely fills $28$ bags with no coffee left over, which of the following equations can be used to find the total weight, $c$ , in pounds, of the coffee batch?

\frac{c}{1.5} = 28

1.5c = 28

c - 1.5 = 28

\frac{1.5}{c} = 28

c + 1.5 = 28

Cevabı ve açıklamayı göster

Cevap: $\frac{c}{1.5} = 28$

Cevap

The equation that can be used to find the total weight is

\frac{c}{1.5} = 28

.

To find the number of bags filled, the total weight of coffee (

c

) must be divided into equal parts of

1.5

pounds each. Setting this quotient equal to the total number of filled bags (

28

) gives the equation

\frac{c}{1.5} = 28

.

Adım Adım Çözüm

1

Identify the given quantities and unknown variable

Total weight =

c

pounds; Weight per bag =

1.5

pounds; Total number of bags =

28

.

Establishing clear variable definitions is necessary before setting up an algebraic model.

2

Formulate the relationship between total quantity, unit rate, and total items

\frac{\text{Total Weight}}{\text{Weight per Bag}} = \text{Total Number of Bags}

Dividing a total quantity by the capacity of one container gives the total number of containers filled.

3

Substitute the known values and variable into the relationship formula

\frac{c}{1.5} = 28

This yields the correct one-step linear equation representing the scenario.

Anahtar Kavram

Writing one-step linear equations from word problems involving rates and equal grouping

Tahmini Süre:1m 0s

Soru 342Soru →

A security code consists of four characters generated in order: a letter, a digit, a letter, and a digit.

- The 1st character is a letter chosen from the 5-element set $\{A, B, C, D, E\}$ .
- The 2nd character is a digit chosen from the 9 non-zero digits $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ .
- The 3rd character is a letter chosen from $\{A, B, C, D, E\}$ , but it cannot be the same as the 1st character.
- The 4th character is an even digit chosen from $\{2, 4, 6, 8\}$ , but it cannot be the same as the 2nd character.

How many different security codes can be formed following these rules?

Cevabı ve açıklamayı göster

Cevap: 640

Cevap

640

The problem requires counting combinations with dependent restrictions, which can be solved by splitting into two mutually exclusive cases. If the 2nd character is an odd digit (5 possibilities), the 4th character has all 4 even digits available, giving 5 × 5 × 4 × 4 = 400 valid codes. If the 2nd character is an even digit (4 possibilities), the 4th character cannot match it, leaving 3 options, giving 5 × 4 × 4 × 3 = 240 valid codes. Summing the two distinct cases gives 400 + 240 = 640.

Adım Adım Çözüm

1

Analyze the conditions for the digit positions (2nd and 4th characters).

The restrictions require dividing the counting into two cases based on the parity of the 2nd character.

The 4th character must be chosen from the 4 even digits {2, 4, 6, 8}. If the 2nd character is an odd digit, it does not reduce the choices for the 4th character. If the 2nd character is an even digit, it reduces the remaining choices for the 4th character from 4 to 3.

2

Calculate total possibilities for Case 1 (2nd character is odd).

5 × 5 × 4 × 4 = 400 codes.

There are 5 options for the 1st letter, 5 odd digits {1, 3, 5, 7, 9} for the 2nd character, 4 remaining letters for the 3rd character, and 4 even digits for the 4th character.

3

Calculate total possibilities for Case 2 (2nd character is even).

5 × 4 × 4 × 3 = 240 codes.

There are 5 options for the 1st letter, 4 even digits {2, 4, 6, 8} for the 2nd character, 4 remaining letters for the 3rd character, and 3 remaining even digits for the 4th character.

4

Sum the results of the two cases.

400 + 240 = 640 security codes.

According to the Addition Principle of counting, the total number of outcomes for mutually exclusive events is the sum of their individual outcomes.

Anahtar Kavram

Fundamental Counting Principle with Case Analysis

Soru 343Soru →

A biology student recorded the mass, in grams, of 6 plant specimens after three weeks of growth: $14$ , $9$ , $22$ , $18$ , $12$ , and $15$ . If a 7th plant specimen is added to the data set, what must its mass be, in grams, so that the mean mass of all 7 specimens equals the median mass of the original 6 specimens?

8.5

11.5

14.5

15.0

50.0

Cevabı ve açıklamayı göster

Cevap: $11.5$

Cevap

The mass of the 7th specimen must be 11.5 grams.

To find the required mass of the 7th specimen, first order the original 6 data points from least to greatest:

9, 12, 14, 15, 18, 22

. The median is the average of the two middle values, 14 and 15, which gives

\frac{14 + 15}{2} = 14.5

grams. The problem states that the mean of all 7 specimens must equal 14.5 grams. The sum of 7 values with a mean of 14.5 is

7 \times 14.5 = 101.5

grams. Since the sum of the first 6 specimens is

9 + 12 + 14 + 15 + 18 + 22 = 90

grams, the mass of the 7th specimen must be

101.5 - 90 = 11.5

grams.

Adım Adım Çözüm

1

Sort the original data set to find its median.

The sorted list of masses is

9, 12, 14, 15, 18, 22

.

Finding the median of a data set requires placing the numbers in ascending order.

2

Calculate the median of the original 6 specimens.

\text{Median} = \frac{14 + 15}{2} = 14.5\text{ grams}

.

For an even number of values, the median is the average of the two middle terms.

3

Determine the required total mass of all 7 specimens.

\text{Required Total Mass} = 7 \times 14.5 = 101.5\text{ grams}

.

Mean is calculated by dividing total sum by count, so

\text{Total Sum} = \text{Mean} \times \text{Count}

.

4

Calculate the sum of the original 6 specimens.

\text{Original Sum} = 9 + 12 + 14 + 15 + 18 + 22 = 90\text{ grams}

.

Summing the existing measurements allows us to find the missing amount.

5

Subtract the original sum from the required total sum.

\text{Mass of 7th Specimen} = 101.5 - 90 = 11.5\text{ grams}

.

The difference between the required total sum and the current total sum gives the value of the new addition.

Anahtar Kavram

Descriptive Statistics: Finding a missing data value given a target mean based on a sample median.

Tahmini Süre:1m 15s

Soru 344Soru →

A commercial printing press produces $420$ color brochures every $15$ minutes. If the press operates continuously at this constant rate, how many hours will it take to produce a total of $5,040$ brochures?

0.8

1.8

3

12

180

Cevabı ve açıklamayı göster

Cevap: $3$

Cevap

3 hours

The correct answer is 3 hours because the printing press produces 28 brochures per minute (

420 \div 15

). In one hour (60 minutes), it produces 1,680 brochures (

28 \times 60

). To complete 5,040 brochures, it takes

5,040 \div 1,680 = 3

hours.

Adım Adım Çözüm

1

Determine the unit rate per minute.

420 \text{ brochures} \div 15 \text{ minutes} = 28 \text{ brochures per minute}

.

Finding the unit rate simplifies conversion to hourly production.

2

Convert the unit rate to an hourly rate.

28 \text{ brochures/minute} \times 60 \text{ minutes/hour} = 1,680 \text{ brochures per hour}

.

The question asks for the total time in hours.

3

Calculate the total hours needed to print 5,040 brochures.

5,040 \text{ brochures} \div 1,680 \text{ brochures/hour} = 3 \text{ hours}

.

Dividing total target quantity by hourly rate yields the required time.

Anahtar Kavram

Unit Rates and Multi-Step Proportional Reasoning

Soru 345Soru →

Four numerical values are given below. Arrange these values in order from least to greatest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from least to greatest is

55\%

,

0.565

,

\frac{4}{7}

, and

\frac{3}{5}

.

Converting each quantity into a decimal format yields

55\% = 0.55

,

0.565 = 0.565

,

\frac{4}{7} \approx 0.5714

, and

\frac{3}{5} = 0.60

. Ordering these decimal values from least to greatest yields

0.55 < 0.565 < 0.5714 < 0.60

, which corresponds to the sequence

55\%, 0.565, \frac{4}{7}, \frac{3}{5}

.

Adım Adım Çözüm

1

Convert the percentage to a decimal.

55\% = \frac{55}{100} = 0.55

Converting all values to decimals provides a uniform scale for direct comparison.

2

Convert the fractions to decimals.

\frac{4}{7} \approx 0.5714

and

\frac{3}{5} = 0.60

Dividing the numerator by the denominator converts each fraction into decimal form.

3

Compare all four decimal values:

0.55

,

0.565

,

0.5714

, and

0.60

.

0.55 < 0.565 < 0.5714 < 0.60

Aligning decimal places shows that

0.5500 < 0.5650 < 0.5714 < 0.6000

.

4

Map the ordered decimal values back to their original forms.

55\% < 0.565 < \frac{4}{7} < \frac{3}{5}

Replacing each decimal with its original representation produces the required sequence.

Anahtar Kavram

Converting fractions, decimals, and percents to a common representation (decimals) to compare and order them.

Soru 346Soru →

A local bakery recorded the number of custom cakes sold per week over a 6-week period: $12$ , $15$ , $8$ , $15$ , $14$ , and $10$ . If the bakery sells $x$ cakes in the 7th week such that the mean number of cakes sold per week for the entire 7-week period is $13$ , what is the median number of cakes sold over the 7-week period?

12

13

14

15

17

Cevabı ve açıklamayı göster

Cevap: $14$

Cevap

The median number of cakes sold over the 7-week period is

14

.

The total sales required for a 7-week average of

13

cakes is

7 \times 13 = 91

. Subtracting the sum of the first 6 weeks (

74

) gives

17

cakes for the 7th week. Arranging the 7 weekly values in order (

8, 10, 12, 14, 15, 15, 17

) shows that the 4th value, which is the median, is

14

.

Adım Adım Çözüm

1

Calculate the sum of the sales for the first 6 weeks.

12 + 15 + 8 + 15 + 14 + 10 = 74

To find the missing 7th week sales, we first need the total sales of the known weeks.

2

Determine the required total sales for a 7-week mean of 13.

7 \times 13 = 91

Mean is total sum divided by the number of data points, so Total = Mean

\times

Count.

3

Find the sales in the 7th week (

x

).

x = 91 - 74 = 17

Subtracting the 6-week sum from the 7-week total yields the 7th week value.

4

Order all 7 data points from least to greatest and identify the middle value.

Ordered list:

8, 10, 12, 14, 15, 15, 17

. The 4th item (median) is

14

.

The median of an odd number of data values is the middle value when arranged in ascending order.

Anahtar Kavram

Descriptive Statistics: Combining mean calculations to solve for missing data and finding the median of an ordered dataset.

Tahmini Süre:1m 0s

Soru 347Soru →

A beverage company formulates a fruit punch by blending apple juice, cranberry juice, and sparkling water in the ratio $5 : 3 : 2$ by volume. In a specific batch, there are $12$ liters of sparkling water. A technician then adds $10$ liters of cranberry juice and an additional amount of apple juice so that the new ratio of apple juice to cranberry juice becomes $3 : 2$ . How many liters of apple juice were added to the batch?

12

15

27

42

22

Cevabı ve açıklamayı göster

Cevap: $12$

Cevap

The technician added 12 liters of apple juice.

In the initial ratio of

5 : 3 : 2

,

2

parts represent

12

liters of sparkling water, meaning each part is

6

liters. Thus, the original mixture contains

30

liters of apple juice and

18

liters of cranberry juice. Adding

10

liters of cranberry juice brings its total to

28

liters. For the new apple juice to cranberry juice ratio to be

3 : 2

, the required total apple juice is

\frac{3}{2} \times 28 = 42

liters. Subtracting the initial

30

liters yields

12

liters of added apple juice.

Adım Adım Çözüm

1

Determine the unit value per part in the original ratio.

Since

2

parts correspond to

12

liters of sparkling water,

1

part =

\frac{12}{2} = 6

liters.

The original ratio

5 : 3 : 2

defines the relative proportions of apple juice, cranberry juice, and sparkling water.

2

Calculate the original volumes of apple juice and cranberry juice.

Apple juice =

5 \times 6 = 30

liters; Cranberry juice =

3 \times 6 = 18

liters.

Multiply each component's ratio value by the single-part volume of

6

liters.

3

Calculate the new volume of cranberry juice after adding 10 liters.

New cranberry juice volume =

18 + 10 = 28

liters.

10

liters of cranberry juice were added to the initial

18

liters.

4

Set up a proportion for the new ratio of apple juice to cranberry juice and solve for the added amount

x

.

\frac{30 + x}{28} = \frac{3}{2} \implies 2(30 + x) = 3(28) \implies 60 + 2x = 84 \implies 2x = 24 \implies x = 12.

The new ratio of total apple juice (

30 + x

) to total cranberry juice (

28

) must equal

3 : 2

.

Anahtar Kavram

Multi-step proportional reasoning and scaling ratios when quantities change.

Tahmini Süre:1m 15s

Soru 348Soru →

A high school cafeteria offers a lunch meal combination consisting of 1 main dish, 1 side dish, and 1 drink. There are 4 main dish options, 5 side dish options, and 3 drink options available. However, due to dietary restrictions, 1 specific main dish option cannot be paired with 2 specific side dish options. How many different valid lunch meal combinations of 1 main dish, 1 side dish, and 1 drink can a student choose?

Cevabı ve açıklamayı göster

Cevap: 54

Cevap

The total number of valid lunch meal combinations is 54.

The total number of unrestricted meal options is

4 \times 5 \times 3 = 60

. The restricted main dish cannot be paired with 2 specific side dishes across any of the 3 drink options, creating

1 \times 2 \times 3 = 6

invalid combinations. Subtracting these from the total gives

60 - 6 = 54

valid combinations.

Adım Adım Çözüm

1

Calculate the total number of unrestricted meal choices

4 \times 5 \times 3 = 60

total combinations

By the Fundamental Counting Principle, multiplying the available choices for each dish component yields the total possible combinations without restrictions.

2

Determine the number of invalid combinations caused by dietary restrictions

1 \times 2 \times 3 = 6

invalid combinations

The 1 restricted main dish cannot be paired with 2 specific side dishes, and each of those 2 pairs can still be matched with any of the 3 drinks.

3

Subtract invalid combinations from the unrestricted total

60 - 6 = 54

valid combinations

Subtracting the disallowed combinations leaves only the combinations that satisfy all dietary restrictions.

Anahtar Kavram

Fundamental Counting Principle with Restrictions

Soru 349Soru →

A contractor prepares a concrete mixture by combining cement, sand, and gravel in a ratio of $2 : 3 : 5$ by weight. The contractor currently has a batch of this mixture weighing $150\text{ pounds}$ . If the contractor adds $10\text{ pounds}$ of pure cement to this batch, what fraction of the total weight of the modified mixture is cement?

\frac{1}{5}

\frac{3}{40}

\frac{1}{4}

\frac{4}{15}

\frac{1}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{4}$

Cevap

The fraction of the total weight of the modified mixture that is cement is

\frac{1}{4}

.

In the original ratio of

2 : 3 : 5

, there are

10

total parts. Cement accounts for

\frac{2}{10} = \frac{1}{5}

of the initial

150\text{-pound}

mixture, which is

30\text{ pounds}

. When

10\text{ pounds}

of cement is added, the new cement weight becomes

40\text{ pounds}

and the total mixture weight increases to

160\text{ pounds}

. The fraction of cement in the modified mixture is

\frac{40}{160} = \frac{1}{4}

.

Adım Adım Çözüm

1

Calculate the total ratio parts and the initial amount of cement in the mixture.

Total ratio parts =

2 + 3 + 5 = 10

. Initial cement weight =

\frac{2}{10} \times 150 = 30\text{ pounds}

.

Determining the weight of cement in the original

150\text{-pound}

batch is necessary before adding extra cement.

2

Determine the new weight of cement and the new total weight after adding

10\text{ pounds}

of cement.

New cement weight =

30 + 10 = 40\text{ pounds}

. New total weight =

150 + 10 = 160\text{ pounds}

.

Adding pure cement increases both the weight of the cement component and the total weight of the batch.

3

Form the part-to-whole ratio for the new mixture and simplify the fraction.

\frac{40}{160} = \frac{1}{4}

.

The fraction of cement relative to the whole mixture is found by dividing the new cement weight by the new total weight.

Anahtar Kavram

Part-to-whole ratios and adjusting totals when components are added

Soru 350Soru →

A music ensemble director must randomly select 3 students from a group consisting of 5 vocalists and 4 instrumentalists to perform at a ceremony. What is the probability that the chosen group contains at least 1 vocalist and at least 1 instrumentalist?

\frac{1}{6}

\frac{10}{21}

\frac{5}{21}

\frac{20}{81}

\frac{5}{6}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{6}$

Cevap

\frac{5}{6}

To find the probability that the selected trio includes at least 1 vocalist and at least 1 instrumentalist, calculate the total possible combinations of picking 3 students from 9:

\binom{9}{3} = 84

. Next, identify the unfavorable combinations where all 3 students are of the same type:

\binom{5}{3} = 10

ways for all vocalists and

\binom{4}{3} = 4

ways for all instrumentalists, totaling

10 + 4 = 14

unfavorable outcomes. Subtracting these from 84 leaves 70 favorable combinations. Thus, the probability is

\frac{70}{84} = \frac{5}{6}

.

Adım Adım Çözüm

1

Calculate the total number of ways to choose any 3 students from the total group of 9 students.

\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84

total outcomes.

Establishes the size of the sample space for selecting 3 students without replacement out of 9 (5 vocalists + 4 instrumentalists).

2

Determine the number of unfavorable outcomes (groups consisting entirely of one type of student).

Ways to choose 3 vocalists:

\binom{5}{3} = 10

.
Ways to choose 3 instrumentalists:

\binom{4}{3} = 4

.
Total unfavorable outcomes =

10 + 4 = 14

.

The complement of having 'at least 1 vocalist and at least 1 instrumentalist' is selecting a group of 3 composed entirely of vocalists or entirely of instrumentalists.

3

Subtract the unfavorable outcomes from the total outcomes to get the favorable outcomes.

84 - 14 = 70

favorable combinations.

Finding the complementary set count is direct and prevents missing valid composition cases.

4

Calculate the probability by dividing favorable outcomes by total outcomes.

Probability =

\frac{70}{84} = \frac{5}{6}

.

Dividing favorable combinations by the total size of the sample space gives the required probability.

Anahtar Kavram

Complementary Probability and Combinations

Tahmini Süre:1m 30s

Soru 351Soru →

A music festival stage technician mounts a series of $n$ identical LED light panels side-by-side along the main stage frame. The main stage frame has a total length of $168$ feet, and each LED light panel spans a length of $3.5$ feet. Which of the following equations can be solved to find $n$ , the total number of LED light panels mounted along the frame?

n + 3.5 = 168

\frac{n}{3.5} = 168

3.5n = 168

n - 3.5 = 168

168n = 3.5

Cevabı ve açıklamayı göster

Cevap: $3.5n = 168$

Cevap

The equation

3.5n = 168

can be solved to find the total number of LED light panels.

The total length of the stage frame is equal to the length of a single panel multiplied by the total number of panels. Since each panel is

3.5

feet long and there are

n

panels, the expression for total length is

3.5n

. Setting this product equal to the total length of

168

feet gives

3.5n = 168

.

Adım Adım Çözüm

1

Identify the relationship between total length, individual length, and quantity.

Total Length = (Length of 1 Panel)

\times

(Number of Panels)

Since all panels are identical and placed end-to-end, their combined length is equal to the total frame length.

2

Substitute given values and variables into the rate relationship.

168 = 3.5 \times n

or

3.5n = 168

Given that length per panel is

3.5

feet, number of panels is

n

, and total frame length is

168

feet.

Anahtar Kavram

Translating real-world multiplicative rate scenarios into one-step linear equations.

Tahmini Süre:1m 0s

Soru 352Soru →

At a local bookstore, $40\%$ of the total inventory consists of fiction books. Of the remaining inventory, $\frac{5}{8}$ consists of non-fiction books, and the rest are children's books. If there are $270$ children's books in the bookstore, what is the total number of books in the inventory?

450

675

720

1,200

1,800

Cevabı ve açıklamayı göster

Cevap: $1,200$

Cevap

1,200

Fiction books account for

40\%

of the total inventory, leaving

60\%

as the remaining portion. Since

\frac{5}{8}

of this remainder is non-fiction, the children's books represent the other

\frac{3}{8}

of that

60\%

remainder. Multiplying

\frac{3}{8} \times 0.60 = 0.225

, or

22.5\%

of the total inventory. Given that there are

270

children's books, dividing

270

by

0.225

yields the total inventory of

1,200

books.

Adım Adım Çözüm

1

Determine the fraction of inventory remaining after accounting for fiction books.

Remaining fraction =

100\% - 40\% = 60\% = 0.6 = \frac{3}{5}

Fiction accounts for

40\%

, leaving

60\%

of the total inventory for other genres.

2

Find the fraction of the remaining inventory that consists of children's books.

Children's fraction of remaining =

1 - \frac{5}{8} = \frac{3}{8}

Since

\frac{5}{8}

of the remaining inventory is non-fiction, the rest (

\frac{3}{8}

) must be children's books.

3

Calculate the proportion of the total inventory that children's books represent.

Children's proportion of total =

\frac{3}{8} \times \frac{3}{5} = \frac{9}{40} = 0.225 = 22.5\%

Multiply the fraction of the remainder by the fraction of the total remaining.

4

Set up an equation with total inventory

T

and solve for

T

.

0.225 \times T = 270 \implies T = \frac{270}{0.225} = 1,200

Divide the known count of children's books (

270

) by its fractional share of the total inventory.

Anahtar Kavram

Solving multi-step word problems involving fractions, decimals, and percentages of a whole

Tahmini Süre:1m 15s

Soru 353Soru →

A digital audio engineer uses a software tool where the total duration $t$ , in seconds, of an electronic music composition is modeled by the equation $t = \frac{b}{2.4}$ , where $b$ represents the total number of beats in the track. If a composition has a total duration of $180$ seconds, what is the total number of beats, $b$ , in the track?

75

177.6

252

432

450

Cevabı ve açıklamayı göster

Cevap: $432$

Cevap

The total number of beats in the track is

432

.

To solve for

b

in the one-step equation

180 = \frac{b}{2.4}

, multiply both sides of the equation by

2.4

. Performing

180 \times 2.4

yields

432

, which is the total number of beats in the track.

Adım Adım Çözüm

1

Substitute the given duration into the formula

180 = \frac{b}{2.4}

The track duration

t

is given as

180

seconds.

2

Isolate the variable

b

by multiplying both sides by

2.4

b = 180 \times 2.4 = 432

Multiplication is the inverse operation of division.

Anahtar Kavram

Solving one-step linear equations involving division by isolating the variable using multiplication.

Tahmini Süre:1m 0s

Soru 354Soru →

A coffee shop blend initially consists of a $20$ -pound mixture of Arabica and Robusta beans, of which $45\%$ by weight is Arabica. The shop manager adds pure Arabica beans to the mixture to increase the proportion of Arabica beans to $56\%$ by weight. How many pounds of pure Arabica beans must be added to the mixture?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5 pounds of pure Arabica beans must be added to the mixture.

To solve for the required added weight, first compute the original mass of Arabica beans:

45\%

of

20

pounds is

0.45 \times 20 = 9

pounds. Letting

x

represent the pounds of pure Arabica added, the updated mass of Arabica is

9 + x

pounds and the updated total mass of the coffee batch is

20 + x

pounds. Setting the ratio

\frac{9 + x}{20 + x} = 0.56

gives

9 + x = 0.56(20 + x)

. Expanding the right side results in

9 + x = 11.2 + 0.56x

. Subtracting

0.56x

and

9

from both sides gives

0.44x = 2.2

, which simplifies to

x = 5

pounds.

Adım Adım Çözüm

1

Determine the initial weight of Arabica beans.

Arabica weight =

0.45 \times 20 = 9

pounds.

The starting

20

-pound batch contains

45\%

Arabica beans.

2

Formulate the mixture fraction with the unknown added weight

x

.

\frac{9 + x}{20 + x} = 0.56

Adding

x

pounds of pure Arabica increases both the total Arabica weight (numerator) and the total mixture weight (denominator).

3

Solve the algebraic equation for

x

.

9 + x = 11.2 + 0.56x \implies 0.44x = 2.2 \implies x = 5

Isolating

x

yields the exact amount of pure Arabica required.

Anahtar Kavram

Solving percentage mixture problems using algebraic proportions

Tahmini Süre:1m 30s

Soru 355Soru →

At a regional debate tournament, Team Alpha has $6$ members with a mean score of $84$ points, and Team Beta has $4$ members with a mean score of $94$ points. If $2$ new members join Team Alpha such that the overall mean score for all $12$ members on both teams increases to $89$ points, what is the mean score of the $2$ new members?

89

90

94

104

188

Cevabı ve açıklamayı göster

Cevap: 94

Cevap

The mean score of the 2 new members is 94.

To find the mean score of the 2 new members, first determine the total points scored by the initial 10 members. Team Alpha scored

6 \times 84 = 504

points, and Team Beta scored

4 \times 94 = 376

points, giving an initial combined total of

880

points. Next, determine the total points required for all 12 members to have a mean score of 89, which is

12 \times 89 = 1068

points. The difference,

1068 - 880 = 188

points, represents the total points scored by the 2 new members. Dividing this sum by 2 yields a mean score of 94 points.

Adım Adım Çözüm

1

Calculate the total initial points for both teams combined.

Team Alpha total =

6 \times 84 = 504

. Team Beta total =

4 \times 94 = 376

. Combined initial total =

504 + 376 = 880

points.

The sum of data values equals the mean multiplied by the number of values.

2

Calculate the required total points for all 12 members to achieve the target overall mean.

Target total points =

12 \times 89 = 1068

points.

The total points required for

12

members with a mean of

89

is

12 \times 89

.

3

Determine the total points contributed by the 2 new members.

Points contributed =

1068 - 880 = 188

points.

Subtracting the initial combined points from the required total points isolates the points added by the

2

new members.

4

Calculate the mean score of the 2 new members.

Mean score =

188 \div 2 = 94

points.

Dividing the total points of the new members by the count of new members (

2

) yields their mean score.

Anahtar Kavram

Weighted Mean and Target Mean Problem Solving

Soru 356Soru →

A water desalination plant processes seawater using two filtration units, Unit P and Unit Q. Unit P can process $720$ gallons of water in $45$ minutes. Unit Q processes water at a rate that is $25\%$ faster than Unit P. If both units operate continuously at their respective rates, how many total gallons of water can Unit P and Unit Q process combined in $2.5$ hours?

3,000

4,200

5,400

8,550

9,000

Cevabı ve açıklamayı göster

Cevap: $5,400$

Cevap

5,400

gallons

Unit P processes water at a rate of

16

gallons per minute (

720 \div 45

). Unit Q operates

25\%

faster, making its rate

20

gallons per minute (

16 \times 1.25

). Working simultaneously, their combined rate is

36

gallons per minute (

16 + 20

). Converting

2.5

hours into

150

minutes (

2.5 \times 60

) and multiplying by the combined rate gives a total of

36 \times 150 = 5,400

gallons.

Adım Adım Çözüm

1

Calculate the unit processing rate of Unit P per minute.

\frac{720 \text{ gallons}}{45 \text{ minutes}} = 16 \text{ gallons per minute}

Divide the total gallons processed by Unit P by the number of minutes to find its unit rate.

2

Determine the unit processing rate of Unit Q per minute.

16 \times (1 + 0.25) = 20 \text{ gallons per minute}

Unit Q is 25% faster than Unit P, so multiply Unit P's rate by 1.25.

3

Calculate the combined processing rate of both units.

16 + 20 = 36 \text{ gallons per minute}

Add the individual rates of Unit P and Unit Q to get their joint processing speed.

4

Convert 2.5 hours to minutes and calculate the combined total gallons processed.

2.5 \text{ hours} \times 60 \text{ minutes/hour} = 150 \text{ minutes}

;

36 \times 150 = 5,400 \text{ gallons}

Multiply the combined rate per minute by the total operating time in minutes.

Anahtar Kavram

Unit rate calculation and proportional scaling across multiple rate sources

Soru 357Soru →

A commercial bakery uses an automated dough divider that cuts a large batch of dough of mass $m$ kilograms into $48$ identical portioned rolls. If each portioned roll has a mass of $0.125$ kilograms, what was the total mass $m$ , in kilograms, of the original batch of dough?

0.0026

6.00

47.875

48.125

384.00

Cevabı ve açıklamayı göster

Cevap: $6.00$

Cevap

The total mass of the original batch of dough is

6.00

kilograms.

The scenario describes dividing a total mass

m

into

48

equal parts, where each part equals

0.125

kg. Setting up the one-step division equation

\frac{m}{48} = 0.125

and multiplying both sides by

48

gives

m = 6.00

kilograms.

Adım Adım Çözüm

1

Define the algebraic relationship based on the problem description.

\frac{m}{48} = 0.125

Dividing the total mass

m

into

48

equal pieces yields the mass of one individual roll.

2

Isolate the variable

m

by performing the inverse operation.

m = 48 \times 0.125

Multiply both sides of the equation by

48

to cancel out the denominator.

3

Calculate the numerical product.

m = 6.00

48 \times 0.125 = 6.00

kilograms.

Anahtar Kavram

Basic Algebraic Expressions and One-Step Equations

Soru 358Soru →

A weather researcher recorded the daily high temperatures, in degrees Fahrenheit, for $5$ consecutive days in a mountain town: $68^\circ\text{F}$ , $74^\circ\text{F}$ , $71^\circ\text{F}$ , $65^\circ\text{F}$ , and $72^\circ\text{F}$ . If the daily high temperature on the 6th day is $x^\circ\text{F}$ , and the mean high temperature for all $6$ days is $72^\circ\text{F}$ , what is the value of $x$ ?

10

72

74

82

88

Cevabı ve açıklamayı göster

Cevap: $82$

Cevap

82

To achieve a target mean of

72^\circ\text{F}

over

6

days, the total sum of all

6

daily high temperatures must be

6 \times 72 = 432^\circ\text{F}

. Adding the

5

recorded temperatures gives

68 + 74 + 71 + 65 + 72 = 350^\circ\text{F}

. Subtracting

350

from

432

yields

82^\circ\text{F}

, which is the temperature needed on the 6th day.

Adım Adım Çözüm

1

Calculate the total sum of the temperatures for the first

5

days.

68 + 74 + 71 + 65 + 72 = 350

To find the missing value, we first need the combined total of the known data points.

2

Determine the required total sum for all

6

days using the target mean.

\text{Total Sum} = 6 \times 72 = 432

The mean of

6

values is equal to the sum of all

6

values divided by

6

.

3

Subtract the

5

-day sum from the total

6

-day sum to solve for

x

.

x = 432 - 350 = 82

The difference between the required total sum and the existing sum gives the value of the 6th data point.

Anahtar Kavram

Finding a missing data value using a target arithmetic mean

Tahmini Süre:1m 0s

Soru 359Soru →

A middle school librarian recorded the number of books checked out by $15$ students during a reading program, as summarized in the table below:

Number of Books	Number of Students
$1$	$3$
$2$	$4$
$3$	$5$
$4$	$2$
$5$	$1$

What is the positive difference between the median and the mean of the number of books checked out per student?

0.0

0.4

0.6

2.4

4.8

Cevabı ve açıklamayı göster

Cevap: $0.4$

Cevap

The positive difference between the median and the mean is

0.4

.

To find the mean, multiply each number of books by the corresponding number of students to find the total sum of books:

(1)(3) + (2)(4) + (3)(5) + (4)(2) + (5)(1) = 39

. Dividing

39

by the total of

15

students yields a mean of

2.6

. To find the median of

15

ordered values, locate the 8th value. The cumulative frequencies show that the 1st through 3rd values are

1

, the 4th through 7th values are

2

, and the 8th through 12th values are

3

. Thus, the median is

3

. The positive difference between the median and mean is

3 - 2.6 = 0.4

.

Adım Adım Çözüm

1

Calculate the total number of books checked out by all students.

Total books =

(1 \times 3) + (2 \times 4) + (3 \times 5) + (4 \times 2) + (5 \times 1) = 3 + 8 + 15 + 8 + 5 = 39

.

Each book quantity must be multiplied by its corresponding student frequency.

2

Calculate the mean number of books per student.

Mean =

\frac{39}{15} = 2.6

.

Divide the total number of books by the total number of students.

3

Determine the median number of books.

The total frequency is

15

, so the median is the

\frac{15 + 1}{2} = 8\text{th}

value. Cumulative frequencies are

3

(for 1 book),

7

(for 2 books), and

12

(for 3 books). The 8th value falls in the 3-book category, so Median =

3

.

The median represents the middle data point when all 15 student values are listed in non-decreasing order.

4

Calculate the positive difference between the median and the mean.

Difference =

3 - 2.6 = 0.4

.

Subtract the mean from the median to find the difference.

Anahtar Kavram

Calculating weighted mean and median from a frequency table.

Tahmini Süre:1m 30s

Soru 360Soru →

A high school athletic department set a total fundraising goal of $\$2,400$ for new equipment. During the first week, the department raised $\frac{3}{8}$ of the total goal. During the second week, they raised $35\%$ of the total goal. During the third week, they raised $0.15$ of the total goal. How much money must the department raise in the fourth week to meet its total goal exactly?

\$288

\$300

\$480

\$615

\$2,100

Cevabı ve açıklamayı göster

Cevap: $\$300$

Cevap

The athletic department must raise

\$\text{300}

in the fourth week to meet its total goal.

Converting each portion into a dollar value relative to the total goal of

\$\text{2,400}

gives

\$\text{900}

for the first week,

\$\text{840}

for the second week, and

\$\text{360}

for the third week. Adding these amounts yields

\$\text{2,100}

raised in total. Subtracting

\$\text{2,100}

from

\$\text{2,400}

gives the remaining requirement of

\$\text{300}

.

Adım Adım Çözüm

1

Calculate the amount raised in Week 1 using the fraction

\frac{3}{8}

.

\frac{3}{8} \times \$2,400 = \$900

To find a fractional part of a total amount, multiply the total by the fraction.

2

Calculate the amount raised in Week 2 using the percentage

35\%

.

0.35 \times \$2,400 = \$840

Convert

35\%

to a decimal (

0.35

) and multiply by the total goal.

3

Calculate the amount raised in Week 3 using the decimal

0.15

.

0.15 \times \$2,400 = \$360

Multiply the decimal fraction directly by the total goal.

4

Sum the amounts raised during the first three weeks and subtract from the total goal.

Total raised =

\$900 + \$840 + \$360 = \$2,100

; Remaining needed =

\$2,400 - \$2,100 = \$300

Subtracting the total funds collected so far from the target yields the remaining requirement.

Anahtar Kavram

Fractions, Decimals, and Percentages

Pre-Algebra

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