Pre-Algebra Alıştırma Soruları — ACT — Sayfa 20

Soru 381Soru →

At an agricultural orchard, a seasonal harvest consisted of apples, pears, and peaches. Of the total harvest, $\frac{3}{8}$ were apples and $42.5\%$ were pears. If the remaining harvest consisted of $96$ bushels of peaches, what was the total number of bushels of fruit harvested by the orchard?

120

256

384

480

640

Cevabı ve açıklamayı göster

Cevap: $480$

Cevap

The total harvest was 480 bushels of fruit.

Converting the fraction of apples

\frac{3}{8}

gives

37.5\%

. Combining this with the

42.5\%

of pears yields

80\%

of the total harvest. The remaining

20\%

of the harvest represents the

96

bushels of peaches. Dividing

96

by

0.20

gives the total harvest of

480

bushels.

Adım Adım Çözüm

1

Convert the fraction of apples to a percentage.

\frac{3}{8} = 0.375 = 37.5\%

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

2

Find the combined percentage of apples and pears.

37.5\% + 42.5\% = 80\%

Adding the percentage of apples and pears determines the portion of the harvest that is not peaches.

3

Calculate the percentage representing peaches.

100\% - 80\% = 20\%

The remaining percentage of the total harvest corresponds to the peaches.

4

Solve for the total number of bushels harvested.

96 \div 0.20 = 480

bushels

Dividing the number of peach bushels by its corresponding decimal fraction (

0.20

) yields the total harvest.

Anahtar Kavram

Solving word problems involving mixed fractions, decimals, percentages, and unknown total quantities.

Tahmini Süre:1m 15s

Soru 382Soru →

A solar power facility generates a total of $E$ kilowatt-hours (kWh) of electricity on a sunny day. This energy is distributed equally into $16$ identical utility-scale battery units, resulting in exactly $425\text{ kWh}$ stored per unit. Which of the following equations can be solved to find $E$ ?

\frac{E}{16} = 425

16E = 425

E - 16 = 425

E + 16 = 425

\frac{16}{E} = 425

Cevabı ve açıklamayı göster

Cevap: $\frac{E}{16} = 425$

Cevap

The equation

\frac{E}{16} = 425

correctly models the relationship.

Because the total energy

E

is divided equally among

16

units, the amount per unit is given by

\frac{E}{16}

. Setting this equal to

425

yields the correct equation

\frac{E}{16} = 425

.

Adım Adım Çözüm

1

Identify the relationship between total quantity, number of groups, and quantity per group.

Total energy

E

divided by the

16

identical units equals the energy per unit (

425\text{ kWh}

).

Equal sharing or distribution corresponds to division in algebraic expressions.

2

Write the one-step algebraic equation reflecting this relationship.

\frac{E}{16} = 425

This sets the quotient of total energy and number of units equal to the single unit amount.

Anahtar Kavram

Formulating one-step division equations from equal distribution scenarios

Soru 383Soru →

A high school photography club has $12$ active members, consisting of $7$ seniors and $5$ juniors. The club needs to elect an executive board composed of a President, a Vice President, and a Secretary, where no member can hold more than one position. If the President must be a senior, how many different executive board arrangements are possible?

Cevabı ve açıklamayı göster

Cevap: 770

Cevap

The total number of different executive board arrangements possible is

770

.

To calculate the total number of distinct executive board arrangements, use the Fundamental Counting Principle by calculating the number of options for each position sequentially. The President must be a senior, so there are 7 choices for President. Once the President is chosen, any of the remaining 11 members can serve as Vice President. After filling the Vice President role, 10 members remain for Secretary. Multiplying these independent choices gives

7 \times 11 \times 10 = 770

.

Adım Adım Çözüm

1

Determine the available choices for the President position

7 possibilities

The President position is restricted to seniors only, and there are 7 seniors in the club.

2

Determine the available choices for the Vice President position

11 possibilities

After 1 member is chosen as President, 11 of the original 12 members remain available for the Vice President role.

3

Determine the available choices for the Secretary position

10 possibilities

After 2 members are chosen for President and Vice President, 10 members remain available for Secretary.

4

Calculate the total number of distinct outcomes using the Fundamental Counting Principle

770 total arrangements

Multiplying the choices for each position gives

7 \times 11 \times 10 = 770

.

Anahtar Kavram

Fundamental Counting Principle with Position Restrictions

Tahmini Süre:1m 15s

Soru 384Soru →

A municipal solar power facility distributes its total generating capacity among three sectors: residential, commercial, and municipal. The residential sector is allocated $\frac{7}{15}$ of the total capacity, and the commercial sector is allocated $36\%$ of the total capacity. The remaining $26$ megawatts (MW) of capacity is allocated to municipal buildings. What is the total generating capacity, in megawatts, of the facility?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The total generating capacity of the facility is 150 megawatts.

The residential portion (

\frac{7}{15}

) and commercial portion (

36\% = \frac{9}{25}

) combine to equal

\frac{62}{75}

of the total capacity. The municipal portion makes up the remaining

\frac{13}{75}

of the total, which is given as

26

MW. Setting

\frac{13}{75} T = 26

and solving yields

T = 150

MW.

Adım Adım Çözüm

1

Convert the commercial percentage into a simplified fraction.

36\% = \frac{36}{100} = \frac{9}{25}

Converting all proportions to fractions allows for direct operations.

2

Find the combined fraction of capacity allocated to residential and commercial sectors.

\frac{7}{15} + \frac{9}{25} = \frac{35}{75} + \frac{27}{75} = \frac{62}{75}

Finding a common denominator (

75

) allows adding the two fractional parts.

3

Find the fraction corresponding to the remaining municipal portion.

1 - \frac{62}{75} = \frac{13}{75}

Subtracting the allocated fraction from

1

yields the unallocated fractional portion.

4

Calculate the total capacity by setting up a linear equation.

\frac{13}{75} \times \text{Total} = 26 \implies \text{Total} = 26 \times \frac{75}{13} = 150

Multiplying the known remaining value by the reciprocal of its fraction gives the whole total.

Anahtar Kavram

Combining fractions and percentages to find an unknown total amount.

Tahmini Süre:1m 30s

Soru 385Soru →

A marine biology research team collected a water sample from an estuary to analyze its composition. In the sample, $\frac{3}{10}$ of the total volume is saltwater, $45\%$ of the total volume is brackish water, and $0.05$ of the total volume is mineral sediment. The remaining portion of the sample is freshwater. What fraction of the total sample volume is freshwater?

\frac{1}{10}

\frac{1}{4}

\frac{1}{5}

\frac{7}{20}

\frac{2}{5}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{5}$

Cevap

The fraction of the total sample volume that is freshwater is

\frac{1}{5}

.

Converting all three non-freshwater portions to decimals yields

0.30

(saltwater),

0.45

(brackish water), and

0.05

(sediment). Adding these together gives

0.80

. Subtracting

0.80

from the total sample volume of

1.00

leaves

0.20

, which converts to the simplified fraction

\frac{1}{5}

.

Adım Adım Çözüm

1

Convert all given volume proportions into decimals.

Saltwater =

\frac{3}{10} = 0.30

, Brackish water =

45\% = 0.45

, Mineral sediment =

0.05

.

Converting all components to a common decimal format allows direct addition.

2

Sum the proportions of saltwater, brackish water, and mineral sediment.

0.30 + 0.45 + 0.05 = 0.80

.

This calculates the combined proportion of the sample that is not freshwater.

3

Subtract the combined non-freshwater proportion from the whole sample (

1.00

).

1.00 - 0.80 = 0.20

.

The remainder of the sample represents the freshwater component.

4

Convert the decimal portion to a simplified fraction.

0.20 = \frac{20}{100} = \frac{1}{5}

.

The question asks for the proportion formatted as a simplified fraction.

Anahtar Kavram

Converting between fractions, decimals, and percentages to solve multi-step word problems.

Soru 386Soru →

A high-speed express train travels a distance of $180\text{ miles}$ in $1.5\text{ hours}$ at a constant speed. For an express schedule, its speed is increased by $20\%$ . At this new constant speed, how many miles will the train travel in $2.5\text{ hours}$ ?

216\text{ miles}

300\text{ miles}

360\text{ miles}

375\text{ miles}

450\text{ miles}

Cevabı ve açıklamayı göster

Cevap: $360\text{ miles}$

Cevap

360\text{ miles}

The original speed is found by dividing

180\text{ miles}

by

1.5\text{ hours}

, which equals

120\text{ miles per hour}

. Increasing this speed by

20\%

yields

120 \times 1.20 = 144\text{ miles per hour}

. Multiplying the new speed of

144\text{ miles per hour}

by

2.5\text{ hours}

gives

360\text{ miles}

.

Adım Adım Çözüm

1

Calculate the original speed of the train in miles per hour.

Original speed =

\frac{180\text{ miles}}{1.5\text{ hours}} = 120\text{ miles per hour}

.

Dividing total distance by total time gives the original unit rate.

2

Calculate the new speed after a

20\%

increase.

New speed =

120 \times (1 + 0.20) = 120 \times 1.20 = 144\text{ miles per hour}

.

Increasing a quantity by

20\%

is equivalent to multiplying it by

1.20

.

3

Calculate the distance traveled at the new speed in

2.5\text{ hours}

.

Distance =

144\text{ miles per hour} \times 2.5\text{ hours} = 360\text{ miles}

.

Multiplying the new rate by the new time yields the total distance.

Anahtar Kavram

Unit Rates and Proportional Scaling

Tahmini Süre:1m 15s

Soru 387Soru →

A specialty bakery allows customers to create a custom dessert box. Each box must contain $1$ base, $1$ flavor, and $2$ different toppings. Customers can choose from $3$ base options, $4$ flavor options, and $6$ topping options. How many different custom dessert boxes can a customer create?

72

144

180

216

360

Cevabı ve açıklamayı göster

Cevap: $180$

Cevap

180

different custom dessert boxes can be created.

The total number of outcomes is found by multiplying the number of choices for each component. For the base, there are

3

choices. For the flavor, there are

4

choices. For the

2

different toppings chosen from

6

, order does not matter, so we calculate the combination

\binom{6}{2} = 15

. Multiplying these choices together gives

3 \times 4 \times 15 = 180

.

Adım Adım Çözüm

1

Determine the number of ways to choose the base and flavor.

There are

3

base options and

4

flavor options.

Each selection is an independent choice.

2

Calculate the number of ways to choose

2

different toppings out of

6

available toppings using combinations.

\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 ways.

The order in which the toppings are selected does not matter, so we use the combination formula

\binom{n}{k} = \frac{n!}{k!(n-k)!}

.

3

Apply the Fundamental Counting Principle to find the total number of unique dessert boxes.

Total combinations =

3 \times 4 \times 15 = 180

.

Multiply the number of ways to make each independent choice.

Anahtar Kavram

Fundamental Counting Principle and Combinations

Tahmini Süre:1m 15s

Soru 388Soru →

A box contains $10$ tiles labeled with the integers from $1$ through $10$ . If $2$ tiles are drawn at random without replacement, how many distinct pairs of tiles (where the order of selection does not matter) have a sum that is an odd number?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The total number of distinct pairs with an odd sum is 25.

To obtain an odd sum when adding two integers, one integer must be odd and the other must be even. In the range

1

through

10

, there are

5

odd integers (

1, 3, 5, 7, 9

) and

5

even integers (

2, 4, 6, 8, 10

). To form a pair with an odd sum, one tile must be selected from the

5

odd tiles and one tile must be selected from the

5

even tiles. By the Fundamental Counting Principle, the number of such distinct pairs is

5 \times 5 = 25

.

Adım Adım Çözüm

1

Determine the parity condition for an odd sum

One tile must be odd and the other must be even

The sum of two integers is odd if and only if one addend is odd and the other addend is even.

2

Count the number of odd and even options

5 odd tiles and 5 even tiles

Among the integers

1

through

10

, the odd numbers are

1, 3, 5, 7, 9

(

5

total) and the even numbers are

2, 4, 6, 8, 10

(

5

total).

3

Apply the Fundamental Counting Principle

5 × 5 = 25 distinct pairs

Selecting one odd tile out of 5 possibilities and one even tile out of 5 possibilities gives

5 \times 5 = 25

distinct unordered pairs.

Anahtar Kavram

Fundamental Counting Principle and Parity of Integers

Tahmini Süre:1m 0s

Soru 389Soru →

A community art center offers $40$ different workshops during a summer session. Among these workshops, $22$ are scheduled in the evening, $18$ are scheduled on weekends, and $8$ are scheduled in both the evening and on weekends. If one workshop is selected at random from the $40$ workshops, what is the probability that it is scheduled in the evening, on a weekend, or both?

\frac{1}{5}

\frac{3}{10}

\frac{3}{5}

\frac{4}{5}

1

Cevabı ve açıklamayı göster

Cevap: $\frac{4}{5}$

Cevap

The correct probability is

\frac{4}{5}

.

To find the probability that a randomly chosen workshop is in the evening, on a weekend, or both, calculate the total number of distinct workshops meeting at least one criterion using the formula

N(\text{Evening} \cup \text{Weekend}) = N(\text{Evening}) + N(\text{Weekend}) - N(\text{Evening} \cap \text{Weekend})

. Substituting the given values gives

22 + 18 - 8 = 32

workshops. Dividing by the total

40

workshops gives

\frac{32}{40}

, which reduces to

\frac{4}{5}

.

Adım Adım Çözüm

1

Identify the given counts for each category

Total workshops =

40

, Evening workshops =

22

, Weekend workshops =

18

, Both =

8

.

Extract the necessary components to apply the inclusion-exclusion principle.

2

Calculate the number of workshops in the evening, on a weekend, or both

Number of favorable workshops =

22 + 18 - 8 = 32

.

Workshops scheduled in both categories are counted twice if evening and weekend counts are added directly, so the intersection must be subtracted once.

3

Compute the probability and simplify the fraction

Probability =

\frac{32}{40} = \frac{4}{5}

.

Divide the number of favorable outcomes by the total sample space size.

Anahtar Kavram

Probability of Combined Events (Principle of Inclusion-Exclusion)

Soru 390Soru →

A student council committee consists of $15$ members: $8$ juniors and $7$ seniors. Among the juniors, $3$ are on the debate team. Among the seniors, $4$ are on the debate team. If one committee member is selected at random, what is the probability that the selected member is a senior or is on the debate team?

\frac{4}{15}

\frac{7}{15}

\frac{2}{3}

\frac{11}{15}

\frac{14}{15}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{3}$

Cevap

The probability that the selected member is a senior or on the debate team is

\frac{2}{3}

.

The total number of committee members is

15

. The event consists of selecting someone who is either a senior or on the debate team. There are

7

total seniors and

7

total debate members (

3

juniors and

4

seniors). Using the principle of inclusion-exclusion, the number of favorable outcomes is

7 + 7 - 4 = 10

. The probability is

\frac{10}{15}

, which simplifies to

\frac{2}{3}

.

Adım Adım Çözüm

1

Identify the total number of members in the sample space.

The total number of members is

15

.

This is given as the total committee size.

2

Determine the number of favorable outcomes for the event.

Number of seniors =

7

. Number of juniors on the debate team =

3

. Total favorable outcomes =

7 + 3 = 10

.

To find members who are a senior OR on the debate team, count all seniors (

7

) plus non-senior debate members (

3

) to avoid double-counting.

3

Calculate and simplify the probability fraction.

\text{Probability} = \frac{10}{15} = \frac{2}{3}

.

Divide the favorable outcomes (

10

) by total outcomes (

15

) and simplify by dividing numerator and denominator by

5

.

Anahtar Kavram

Probability of Compound Events (Inclusion-Exclusion Principle)

Soru 391Soru →

A sports analyst tracks four basketball players' shooting efficiencies over a season, expressed in different numerical formats:

- Player P: $\frac{9}{16}$
- Player Q: $0.58$
- Player R: $54.5\%$
- Player S: $\frac{11}{20}$

Place the players in order of their shooting efficiency from least to greatest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

Player R (

54.5\%

), Player S (

\frac{11}{20}

), Player P (

\frac{9}{16}

), Player Q (

0.58

)

To order the shooting efficiencies from least to greatest, convert each representation into a decimal: Player R is

54.5\% = 0.545

, Player S is

\frac{11}{20} = 0.55

, Player P is

\frac{9}{16} = 0.5625

, and Player Q is

0.58

. Comparing these decimals gives

0.545 < 0.55 < 0.5625 < 0.58

, corresponding to the sequence: Player R, Player S, Player P, Player Q.

Adım Adım Çözüm

1

Convert all given values into a common format, such as decimals.

Player P:

\frac{9}{16} = 0.5625

; Player Q:

0.58

; Player R:

54.5\% = 0.545

; Player S:

\frac{11}{20} = 0.55

Converting fractions and percentages into decimals allows for direct numerical comparison.

2

Compare the resulting decimal values.

0.545 < 0.55 < 0.5625 < 0.58

Aligning decimals by place value confirms the ascending sequence.

3

Match the ordered decimals back to the original player designations.

Player R (

0.545

) < Player S (

0.55

) < Player P (

0.5625

) < Player Q (

0.58

)

This establishes the correct order from least to greatest efficiency.

Anahtar Kavram

Comparing and ordering numbers by converting fractions, decimals, and percentages into a unified numerical representation.

Soru 392Soru →

A container holds $6$ red tiles, $4$ blue tiles, and $5$ green tiles. If $2$ tiles are selected at random from the container one after another without replacement, what is the probability that both selected tiles are blue?

\frac{7}{29}

\frac{2}{35}

\frac{6}{55}

\frac{16}{225}

\frac{101}{210}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{35}$

Cevap

The probability that both selected tiles are blue is

\frac{2}{35}

.

The correct answer is determined by calculating the probability of each draw sequentially. The probability of drawing a blue tile first is

\frac{4}{15}

. Because the tile is not replaced,

3

blue tiles and

14

total tiles remain for the second draw, giving a probability of

\frac{3}{14}

. Multiplying these probabilities yields

\frac{4}{15} \times \frac{3}{14} = \frac{12}{210}

, which simplifies to

\frac{2}{35}

.

Adım Adım Çözüm

1

Find the total number of tiles in the container.

Total tiles =

6 + 4 + 5 = 15

.

The sample space for the first selection consists of all available tiles.

2

Calculate the probability that the first tile selected is blue.

Probability of first blue tile =

\frac{4}{15}

.

There are

4

blue tiles out of a total of

15

tiles.

3

Determine the remaining number of blue tiles and total tiles for the second selection.

Remaining blue tiles =

3

, remaining total tiles =

14

.

Since the first tile was selected without replacement, both the count of blue tiles and total tiles decrease by

1

.

4

Calculate the probability that the second tile selected is blue given the first was blue.

Probability of second blue tile =

\frac{3}{14}

.

There are now

3

blue tiles remaining out of

14

total remaining tiles.

5

Multiply the probabilities of the sequential dependent events.

\frac{4}{15} \times \frac{3}{14} = \frac{12}{210} = \frac{2}{35}

.

By the multiplication rule for probability,

P(\text{both blue}) = P(\text{1st blue}) \times P(\text{2nd blue | 1st blue})

.

Anahtar Kavram

Basic Probability of Dependent Sequential Events (Without Replacement)

Daha Fazla Pratik

Try solving a similar question where 3 tiles are selected sequentially without replacement.

Alternatif Yöntem

Calculate using combinations: The total ways to choose any 2 tiles out of 15 is

\binom{15}{2} = \frac{15 \times 14}{2} = 105

. The total ways to choose 2 blue tiles out of 4 is

\binom{4}{2} = \frac{4 \times 3}{2} = 6

. The probability is

\frac{6}{105} = \frac{2}{35}

.

Tahmini Süre:1m 0s

Soru 393Soru →

A regional sports club assigns identification codes to all of its members. Each code consists of $1$ letter chosen from the set $\{K, L, M, N\}$ , followed by $2$ digits chosen from $\{1, 2, 3, 4, 5\}$ such that no digit is repeated within a code, followed by $1$ symbol chosen from $\{*, \#\}$ . How many unique identification codes can be created using this system?

Cevabı ve açıklamayı göster

Cevap: 160

Cevap

The total number of unique identification codes that can be created is

160

.

According to the Fundamental Counting Principle, to find the total number of multi-stage outcomes, multiply the number of choices at each stage. For the letter slot, there are

4

choices. For the two digit slots without repetition, there are

5 \times 4 = 20

choices. For the symbol slot, there are

2

choices. Multiplying these gives

4 \times 20 \times 2 = 160

unique identification codes.

Adım Adım Çözüm

1

Determine the number of available letter choices for the first slot.

4 options

The set of allowed letters

\{K, L, M, N\}

contains 4 distinct elements.

2

Calculate the number of permutations for the two-digit section without repetition.

20 options

Choosing 2 distinct digits from 5 options gives

5 \times 4 = 20

possible outcomes.

3

Determine the number of available symbol choices for the last slot.

2 options

The set of allowed symbols

\{*, \#\}

contains 2 elements.

4

Multiply the number of choices for each slot using the Fundamental Counting Principle.

160 unique codes

Total codes =

4 \times 20 \times 2 = 160

.

Anahtar Kavram

Fundamental Counting Principle and Permutations without Repetition

Tahmini Süre:1m 0s

Soru 394Soru →

A chemistry lab technician measures the concentration of sugar by weight in four distinct liquid samples:

• Sample P: $\frac{7}{16}$
• Sample Q: $0.425$
• Sample R: $44\%$
• Sample S: $\frac{4}{9}$

Which of the following lists the four samples in order from least sugar concentration to greatest sugar concentration?

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from least to greatest concentration is Sample Q, Sample P, Sample R, and Sample S.

To arrange the values from least to greatest, convert all numbers to decimals: Sample Q is

0.425

; Sample P is

\frac{7}{16} = 0.4375

; Sample R is

44\% = 0.44

; and Sample S is

\frac{4}{9} = 0.4444\dots

. Comparing these decimals gives

0.4250 < 0.4375 < 0.4400 < 0.4444\dots

, which corresponds to the sequence Sample Q, Sample P, Sample R, Sample S.

Adım Adım Çözüm

1

Convert each fraction, decimal, and percentage to a common decimal representation.

Sample P = 7/16 = 0.4375; Sample Q = 0.425; Sample R = 44% = 0.44; Sample S = 4/9 ≈ 0.4444...

Converting all numbers to decimals allows for direct place-value comparison.

2

Compare the converted decimal values from left to right.

0.4250 < 0.4375 < 0.4400 < 0.4444...

Comparing digits at the hundredths and thousandths places establishes the relative sizes.

3

Map the ordered decimal values back to their original sample names.

Sample Q < Sample P < Sample R < Sample S

Ordering matches the decimal values calculated: 0.425 < 0.4375 < 0.44 < 0.4444...

Anahtar Kavram

Converting mixed numerical forms (fractions, decimals, percentages) to a single decimal format for ordering.

Soru 395Soru →

A community garden allocates space among three categories: vegetable beds, herb gardens, and flower borders. Vegetable beds take up $\frac{3}{8}$ of the total area, herb gardens take up $0.25$ of the total area, and flower borders occupy the remaining 1,800 square feet. What is the total area, in square feet, of the community garden?

2,700

2,880

4,800

675

7,200

Cevabı ve açıklamayı göster

Cevap: 4,800

Cevap

4,800 square feet

To find the total area, first convert

0.25

to

\frac{2}{8}

. Summing the vegetable portion (

\frac{3}{8}

) and the herb portion (

\frac{2}{8}

) gives

\frac{5}{8}

of the garden. The flower section takes up the remaining

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

of the area. Setting

\frac{3}{8}

of the total area equal to 1,800 square feet and dividing 1,800 by

\frac{3}{8}

gives

1,800 \times \frac{8}{3} = 4,800

square feet.

Adım Adım Çözüm

1

Convert the decimal portion to a fraction with a common denominator.

0.25 = \frac{1}{4} = \frac{2}{8}

Converting all given portions into fractions with a common denominator allows direct addition.

2

Calculate the combined fraction for vegetable beds and herb gardens.

\frac{3}{8} + \frac{2}{8} = \frac{5}{8}

Adding the two known fractions identifies the combined portion of the garden accounted for by vegetables and herbs.

3

Find the fraction representing the flower borders.

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

Subtracting the combined fraction from the whole (1) yields the fractional part representing the remaining 1,800 square feet.

4

Solve for the total garden area.

\text{Total Area} = 1,800 \div \frac{3}{8} = 1,800 \times \frac{8}{3} = 4,800\text{ sq ft}

Dividing the part by its corresponding fractional portion calculates the whole quantity.

Anahtar Kavram

Combining fractions and decimals to solve part-to-whole problems

Tahmini Süre:1m 30s

Soru 396Soru →

A coffee roasting company creates a custom espresso blend using three types of coffee beans: Arabica, Robusta, and Liberica. In the blend, $\frac{3}{8}$ of the total weight is Arabica beans and $40\%$ of the total weight is Robusta beans. The remaining weight of the blend consists of Liberica beans. If a batch of this custom blend contains $18$ ounces of Liberica beans, what is the total weight, in ounces, of the coffee bean batch?

30

48

62

72

80

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

The total weight of the coffee bean batch is 80 ounces.

To find the total weight of the batch, first express each given portion as a decimal or percentage of the total weight. The Arabica beans make up

\frac{3}{8} = 0.375

or

37.5\%

of the total weight, and Robusta beans make up

40\%

of the total weight. Together, Arabica and Robusta represent

37.5\% + 40\% = 77.5\%

of the total weight. Consequently, Liberica beans represent the remaining

100\% - 77.5\% = 22.5\%

(or

0.225

) of the batch. Since

18

ounces equals

22.5\%

of the total weight

W

, we set up the equation

0.225 W = 18

and solve for

W

:

W = \frac{18}{0.225} = 80

ounces.

Adım Adım Çözüm

1

Convert the fractional share of Arabica beans into a decimal or percentage.

\frac{3}{8} = 0.375 = 37.5\%

Converting all components to a common format (percentages or decimals) allows easy addition of the component shares.

2

Find the combined percentage share of Arabica and Robusta beans.

37.5\% + 40\% = 77.5\%

Adding the individual percentages determines the total fraction of the batch occupied by Arabica and Robusta.

3

Calculate the percentage share representing Liberica beans.

100\% - 77.5\% = 22.5\%

(or

0.225

in decimal form)

The remaining percentage of the batch must equal the Liberica portion.

4

Set up and solve the equation to find the total batch weight

W

.

0.225 \times W = 18 \implies W = \frac{18}{0.225} = 80

ounces

Dividing the given weight of Liberica beans by its fractional representation yields the total weight of the entire coffee blend.

Anahtar Kavram

Combining fractions and percentages to solve part-to-whole word problems

Tahmini Süre:1m 30s

Soru 397Soru →

A catering service allows customers to build a custom buffet meal by selecting $1$ appetizer from $4$ available options, $1$ main course from $5$ available options, and $2$ distinct side dishes from $6$ available options. However, due to preparation constraints, $2$ specific side dishes cannot be selected together. How many different valid buffet meal combinations can a customer choose?

23

120

280

300

600

Cevabı ve açıklamayı göster

Cevap: $280$

Cevap

The total number of valid buffet meal combinations is

280

.

There are

4

appetizer choices and

5

main course choices. Choosing

2

side dishes out of

6

options gives

\binom{6}{2} = 15

total pairs. Subtracting the

1

disallowed pair leaves

14

valid side dish pairs. Multiplying these independent choices gives

4 \times 5 \times 14 = 280

valid buffet menu combinations.

Adım Adım Çözüm

1

Calculate the total number of ways to select 2 distinct side dishes from 6 options.

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

combinations.

Since the order in which the 2 side dishes are picked does not matter, combinations are used.

2

Subtract the restricted pair of side dishes from the total side dish combinations.

15 - 1 = 14

valid side dish combinations.

The problem specifies that 2 specific side dishes cannot be selected together, eliminating 1 specific pair.

3

Apply the Fundamental Counting Principle to find the overall number of valid buffet menu choices.

4 \text{ (appetizers)} \times 5 \text{ (main courses)} \times 14 \text{ (valid side pairs)} = 280

.

The choices for appetizer, main course, and side dish pair are independent, so their individual number of possibilities are multiplied.

Anahtar Kavram

Fundamental Counting Principle and Combinations with Restrictions

Tahmini Süre:1m 15s

Soru 398Soru →

In July, a company's IT department logged a set of service tickets. Of these tickets, $\frac{3}{10}$ were categorized as hardware issues, $45\%$ were categorized as software issues, and the remaining 60 tickets were categorized as network issues. What was the total number of service tickets logged by the IT department in July?

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

The total number of service tickets logged in July was 240.

Converting

\frac{3}{10}

gives

30\%

. Combining hardware (

30\%

) and software (

45\%

) issues yields

75\%

of the total tickets. The network tickets make up the remaining

25\%

(

100\% - 75\%

). Setting

25\%

of the total equal to 60 tickets gives a total of

\frac{60}{0.25} = 240

service tickets.

Adım Adım Çözüm

1

Convert the fraction of hardware tickets to a percentage

\frac{3}{10} = 0.30 = 30\%

Expressing all portions as percentages enables direct combination.

2

Sum the percentages for hardware and software tickets

30\% + 45\% = 75\%

This determines the combined percentage of the non-network tickets.

3

Determine the remaining percentage corresponding to network tickets

100\% - 75\% = 25\%

The total of all categories must sum to 100%.

4

Calculate the total number of service tickets

\frac{60}{0.25} = 240

Dividing the quantity of network tickets by their decimal equivalent of 0.25 gives the total count.

Anahtar Kavram

Combining fractions and percentages to solve for an unknown whole amount

Tahmini Süre:1m 30s

Soru 399Soru →

A fair spinner is divided into $8$ congruent sectors numbered $1$ through $8$ . A player spins the spinner twice in succession. How many of the $64$ possible outcomes result in a sum of the two spins that is strictly greater than $12$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

There are 10 outcomes that yield a sum strictly greater than 12.

Systematically listing all ordered pairs

(x, y)

from

\{1, 2, \dots, 8\} \times \{1, 2, \dots, 8\}

such that

x + y > 12

yields 4 outcomes for sum 13, 3 outcomes for sum 14, 2 outcomes for sum 15, and 1 outcome for sum 16, totaling 10 valid outcomes.

Adım Adım Çözüm

1

Determine the acceptable sums for the two spins

The possible sums strictly greater than 12 are 13, 14, 15, and 16.

Since each spin has a maximum value of 8, the maximum possible sum is 8 + 8 = 16.

2

Count the ordered pairs (spin 1, spin 2) for each valid sum

4 outcomes for sum 13, 3 outcomes for sum 14, 2 outcomes for sum 15, and 1 outcome for sum 16.

First and second spins are ordered, so (5,8) and (8,5) represent distinct outcomes.

3

Sum the outcome counts across all valid cases

4 + 3 + 2 + 1 = 10 outcomes.

The sets of outcomes for distinct sums are mutually exclusive.

Anahtar Kavram

Basic Probability and Counting Sample Space Outcomes

Tahmini Süre:1m 15s

Soru 400Soru →

A fitness center surveyed its members regarding their primary exercise routine. Of the members surveyed, $\frac{1}{4}$ chose strength training, $0.35$ chose cardio workouts, and $15\%$ chose group fitness classes. The remaining $75$ members chose swimming. What is the total number of members surveyed by the fitness center?

100

200

250

300

500

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

300

Converting

\frac{1}{4}

to

25\%

and

0.35

to

35\%

, the three specified groups represent

25\% + 35\% + 15\% = 75\%

of the total members. Consequently, the remaining

25\%

of members choose swimming. Since

25\%

(or

\frac{1}{4}

) of the total surveyed group equals 75 members, multiplying 75 by 4 gives a total of 300 members.

Adım Adım Çözüm

1

Convert all given proportions to a common format (percentages or decimals)

Strength training =

\frac{1}{4} = 25\%

; Cardio workouts =

0.35 = 35\%

; Group fitness =

15\%

Converting all amounts to percentages allows for direct addition.

2

Calculate the total percentage accounted for by the first three categories

25\% + 35\% + 15\% = 75\%

Combining these parts determines what fraction of the total surveyed group is already accounted for.

3

Determine the percentage corresponding to the remaining members who chose swimming

100\% - 75\% = 25\%

The remaining members represent the remaining portion of the whole.

4

Set up an equation to find the total number of members (

T

)

0.25 T = 75 \implies T = \frac{75}{0.25} = 300

Dividing the part by its corresponding decimal percentage yields the total population.

Anahtar Kavram

Converting between fractions, decimals, and percentages to calculate an unknown total quantity

Tahmini Süre:1m 15s

Pre-Algebra

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