Pre-Algebra Alıştırma Soruları — ACT — Sayfa 9

What is the value of the expression below?

\frac{\sqrt[3]{2.7 \times 10^7} \cdot \sqrt{1.6 \times 10^9}}{\sqrt{2.5 \times 10^{-1}}}

2.4 \times 10^6

6.0 \times 10^6

1.2 \times 10^7

2.4 \times 10^7

2.4 \times 10^9

Cevabı ve açıklamayı göster

Cevap: $2.4 \times 10^7$

Cevap

The correct value of the expression is

2.4 \times 10^7

.

Evaluating the terms step-by-step, the first term in the numerator simplifies to

3 \times 10^2

and the second term simplifies to

4 \times 10^4

. Multiplying these values results in

1.2 \times 10^7

. The denominator evaluates to

0.5

. Dividing

1.2 \times 10^7

by

0.5

yields

2.4 \times 10^7

.

Adım Adım Çözüm

1

Simplify the first term in the numerator:

\sqrt[3]{2.7 \times 10^7}

.

3 \times 10^2

Rewrite

2.7 \times 10^7

as

27 \times 10^6

. Since 27 is a perfect cube (

3^3

) and

10^6

is a perfect cube (

(10^2)^3

), the cube root is

3 \times 10^2

.

2

Simplify the second term in the numerator:

\sqrt{1.6 \times 10^9}

.

4 \times 10^4

Rewrite

1.6 \times 10^9

as

16 \times 10^8

. Since 16 is a perfect square (

4^2

) and

10^8

is a perfect square (

(10^4)^2

), the square root is

4 \times 10^4

.

3

Multiply the simplified numerator terms.

1.2 \times 10^7

Multiply the coefficients and add the exponents:

(3 \times 10^2) \times (4 \times 10^4) = (3 \times 4) \times 10^{2+4} = 12 \times 10^6 = 1.2 \times 10^7

.

4

Simplify the denominator:

\sqrt{2.5 \times 10^{-1}}

.

5 \times 10^{-1}

(or

0.5

)

Rewrite

2.5 \times 10^{-1}

as

0.25

, or as

25 \times 10^{-2}

. The square root is

\sqrt{25} \times \sqrt{10^{-2}} = 5 \times 10^{-1} = 0.5

.

5

Divide the numerator by the denominator.

2.4 \times 10^7

Divide

1.2 \times 10^7

by

0.5

to get

2.4 \times 10^7

. In scientific notation terms:

\frac{1.2 \times 10^7}{5 \times 10^{-1}} = 0.24 \times 10^8 = 2.4 \times 10^7

.

Anahtar Kavram

Evaluating calculations involving square and cube roots of scientific notation expressions by adjusting decimal places to construct perfect powers.

Soru 162Soru →

A baker makes a fruit tart using strawberries and blueberries. The ratio of the number of strawberries to the number of blueberries used in the tart is $3:5$ . If the baker uses a total of $40$ berries to make the tart, how many strawberries are used?

8

15

20

24

25

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

To find the number of strawberries used, we establish the part-to-whole fraction for strawberries. Given the ratio of strawberries to blueberries is

3:5

, strawberries account for

3

out of every

8

berries (since

3 + 5 = 8

). We then scale this fraction to the total of

40

berries:

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

. This shows that exactly 15 strawberries were used.

Adım Adım Çözüm

1

Determine the part-to-whole ratio for the strawberries.

The ratio of strawberries to blueberries is

3:5

, which means there are

3

parts strawberries for every

3 + 5 = 8

parts of total berries. The fraction of strawberries is

\frac{3}{8}

.

To find the quantity of a specific part from a total, we must use the part-to-whole ratio.

2

Multiply the part-to-whole fraction by the total number of berries.

\frac{3}{8} \times 40 = 3 \times 5 = 15

.

Multiplying the strawberry fraction by the total quantity yields the absolute number of strawberries used.

Anahtar Kavram

Part-to-whole ratios and scaling proportions

Tahmini Süre:45s

Soru 163Soru →

What is the sum of all integer values of $y$ that satisfy the inequality $|2y - 3| \le 7$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The sum of all integer values of

y

that satisfy the inequality is 12.

Solving the inequality

|2y - 3| \le 7

requires setting up the compound inequality

-7 \le 2y - 3 \le 7

. Adding 3 to all parts gives

-4 \le 2y \le 10

, and dividing by 2 yields the interval

-2 \le y \le 5

. The integers in this closed interval are

-2, -1, 0, 1, 2, 3, 4

, and

5

. Summing these values gives 12, as the terms

-2

and

-1

cancel out with

2

and

1

.

Adım Adım Çözüm

1

Set up the compound inequality

-7 \le 2y - 3 \le 7

An absolute value inequality of the form

|a| \le b

translates to

-b \le a \le b

.

2

Isolate the term containing

y

-4 \le 2y \le 10

Add 3 to all three parts of the compound inequality to eliminate the

-3

.

3

Solve for

y

-2 \le y \le 5

Divide all three parts of the inequality by 2.

4

Identify the integer solutions in the interval

-2, -1, 0, 1, 2, 3, 4, 5

The inequality includes the endpoints, so the integers satisfying the inequality are all integers from

-2

through

5

, inclusive.

5

Calculate the sum of the integers

12

Adding the integers:

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 = 12

. The negative integers cancel out their corresponding positive counterparts (

-2

and

2

,

-1

and

1

).

Anahtar Kavram

Solving compound inequalities derived from absolute value inequalities and finding the sum of the integer solution set.

Tahmini Süre:1m 30s

Soru 164Soru →

A company allocates its annual budget to three departments: Research, Marketing, and Operations. Initially, the Research department receives $\frac{1}{4}$ of the total budget. Marketing receives $40\%$ of the remaining budget, and Operations receives the rest. Mid-year, the budget is adjusted: Research's budget is decreased by $16\%$ of its initial allocation, and Marketing's budget is increased by $\frac{1}{12}$ of its initial allocation. If the total company budget remains unchanged, by what percentage must the Operations department's budget increase relative to its initial allocation?

1.5\%

3\frac{1}{3}\%

5\frac{1}{3}\%

7\frac{2}{3}\%

14\frac{4}{9}\%

Cevabı ve açıklamayı göster

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

Cevap

The Operations department's budget must increase by

3\frac{1}{3}\%

relative to its initial allocation.

The correct answer is

3\frac{1}{3}\%

. Under a total budget of

1,000

units, Research initially receives

250

units, Marketing receives

300

units (

40\%

of the remaining

750

), and Operations receives the remaining

450

units. The mid-year decrease for Research is

40

units (

16\%

of

250

), and the increase for Marketing is

25

units (

\frac{1}{12}

of

300

). To keep the total budget constant, the Operations budget must increase by

15

units (

40 - 25

). Relative to its initial budget of

450

units, this represents an increase of

\frac{15}{450} \times 100\% = 3\frac{1}{3}\%

.

Adım Adım Çözüm

1

Set up a total initial budget and find initial allocations for each department.

Let the total initial budget be

1,000

units. Research receives

\frac{1}{4}

of the budget, which is

250

units. The remaining budget is

1,000 - 250 = 750

units. Marketing receives

40\%

of this remaining budget, which is

0.40 \times 750 = 300

units. Operations receives the rest, which is

1,000 - 250 - 300 = 450

units.

Establishing the initial values of each department's budget provides a base to calculate subsequent changes.

2

Calculate the mid-year budget changes for Research and Marketing.

Research decreases by

16\%

:

-0.16 \times 250 = -40

units. Marketing increases by

\frac{1}{12}

:

+\frac{1}{12} \times 300 = +25

units.

Finding the individual change values shows the net budget surplus or deficit before adjusting Operations.

3

Determine the required budget change for the Operations department.

To keep the total budget unchanged, the net sum of changes must be zero:

-40 + 25 + \Delta O = 0 \Rightarrow \Delta O = +15

units.

The change in the Operations budget must balance out the changes in the other two departments.

4

Calculate the percentage increase for the Operations department relative to its initial budget.

Percentage increase =

\frac{15}{450} \times 100\% = \frac{1}{30} \times 100\% = \frac{10}{3}\% = 3\frac{1}{3}\%

.

Dividing the change in Operations by its initial budget yields the relative percentage increase.

Anahtar Kavram

Calculating percentage changes across weighted parts of a whole

Soru 165Soru →

Four investment options, each with a different starting value (expressed as a fraction, decimal, or percentage of a standard initial budget $B$ ) and a specified first-year growth rate, are described below. Arrange the options in order from the lowest final value to the highest final value after the first year.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

Option Y, Option Z, Option X, Option W

The correct order is Option Y, Option Z, Option X, Option W. Converting each initial value to a decimal fraction of

B

and applying the respective growth multipliers yields: Option Y is

0.90 \times 1.15 = 1.035B

, Option Z is

0.875 \times 1.184 = 1.036B

, Option X is

0.85 \times 1.22 = 1.037B

, and Option W is

0.80 \times 1.30 = 1.040B

. Comparing these decimal values gives

1.035 < 1.036 < 1.037 < 1.040

.

Adım Adım Çözüm

1

Calculate the final value of Option W relative to

B

.

0.80 \times 1.30 \times B = 1.040B

Convert the starting fraction

\frac{4}{5}

to the decimal

0.80

. A

30\%

increase corresponds to a multiplier of

1 + 0.30 = 1.30

. Multiply these values together:

0.80 \times 1.30 = 1.040

.

2

Calculate the final value of Option X relative to

B

.

0.85 \times 1.22 \times B = 1.037B

The starting decimal is

0.85

. A

22\%

increase corresponds to a multiplier of

1 + 0.22 = 1.22

. Multiply these values together:

0.85 \times 1.22 = 1.037

.

3

Calculate the final value of Option Y relative to

B

.

0.90 \times 1.15 \times B = 1.035B

Convert the starting percentage

90\%

to the decimal

0.90

. A

15\%

increase corresponds to a multiplier of

1 + 0.15 = 1.15

. Multiply these values together:

0.90 \times 1.15 = 1.035

.

4

Calculate the final value of Option Z relative to

B

.

0.875 \times 1.184 \times B = 1.036B

Convert the starting fraction

\frac{7}{8}

to the decimal

0.875

. An

18.4\%

increase corresponds to a multiplier of

1 + 0.184 = 1.184

. Multiply these values together:

0.875 \times 1.184 = 1.036

.

5

Compare the resulting multipliers to order the final values from lowest to highest.

1.035 < 1.036 < 1.037 < 1.040

, which translates to Option Y < Option Z < Option X < Option W.

By arranging the computed multipliers in ascending order, we find the correct sequence.

Anahtar Kavram

Converting fractions, decimals, and percentages to common decimal representations to compute and compare multi-step percentage increases.

Soru 166Soru →

A rectangular courtyard measures 84 feet by 120 feet. A contractor wants to pave the courtyard using the fewest possible identical square stone tiles such that no tiles are cut and the entire courtyard is covered. What is the total number of square tiles the contractor will need?

12

17

70

280

840

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

70

To find the fewest number of identical square tiles that can cover the courtyard without being cut, we must find the largest possible size for the square tiles. The side length of the square tiles must be a common factor of the dimensions of the courtyard, 84 and 120. To minimize the number of tiles, we maximize the tile size by finding the greatest common factor (GCF) of 84 and 120.

The prime factorization of 84 is

2^2 \times 3 \times 7

.
The prime factorization of 120 is

2^3 \times 3 \times 5

.

The GCF is

2^2 \times 3 = 12

.

Thus, each square tile has a side length of 12 feet. Next, we determine how many tiles fit along each dimension of the courtyard:
Along the 84-foot side:

84 \div 12 = 7

tiles.
Along the 120-foot side:

120 \div 12 = 10

tiles.

The total number of square tiles needed is the product of these two quantities:

7 \times 10 = 70

tiles.

Adım Adım Çözüm

1

Find the greatest common factor (GCF) of the two dimensions, 84 and 120, to determine the maximum side length of the square tiles.

The GCF of 84 and 120 is 12, meaning each square tile will have a side length of 12 feet.

To cover the courtyard with the fewest possible square tiles without cutting, the tiles must be as large as possible, so their side length must be the greatest common divisor of the courtyard's dimensions.

2

Calculate the number of tiles needed along the length and the width of the courtyard.

7 tiles fit along the 84-foot side (

84 \div 12 = 7

), and 10 tiles fit along the 120-foot side (

120 \div 12 = 10

).

This determines the grid dimensions of the tiles covering the rectangular floor.

3

Multiply the number of tiles along the length by the number of tiles along the width to find the total number of tiles.

7 \times 10 = 70

tiles.

The total number of tiles is the product of the grid dimensions.

Anahtar Kavram

Using the Greatest Common Factor (GCF) to solve real-world optimization and division problems.

Tahmini Süre:1m 30s

Soru 167Soru →

On a standard number line, the coordinate of point $P$ is an integer $p$ , and the coordinate of point $Q$ is an integer $q$ . The distance between $P$ and the origin is less than 5, and the distance between $Q$ and $-3$ is exactly 4. If the product $p \cdot q$ is minimized, what is the value of $|p - q|$ ?

3

5

8

11

12

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

|p - q|

is 11, which corresponds to the option with a value of 11.

The distance between

P

and the origin is less than 5, so the integer coordinate

p

must satisfy

|p| < 5

, meaning

p \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}

. The distance between

Q

and

-3

is exactly 4, so

|q - (-3)| = 4 \implies |q + 3| = 4

, which gives

q = 1

or

q = -7

. To minimize the product

p \cdot q

, we analyze the two possibilities for

q

. If

q = 1

, the minimum product is

-4

when

p = -4

. If

q = -7

, the minimum product is

-28

when

p = 4

. The absolute minimum product is

-28

, achieved when

p = 4

and

q = -7

. The value of

|p - q|

for this pair is

|4 - (-7)| = 11

.

Adım Adım Çözüm

1

Determine the possible integer coordinates for point

P

.

p \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}

The distance from

P

to the origin is less than 5, meaning

|p| < 5

. Since

p

is an integer, it can be any integer strictly between

-5

and

5

.

2

Determine the possible integer coordinates for point

Q

.

q = 1

or

q = -7

The distance from

Q

to

-3

is exactly 4, meaning

|q - (-3)| = 4 \implies |q + 3| = 4

. Solving this gives

q + 3 = 4 \implies q = 1

, or

q + 3 = -4 \implies q = -7

.

3

Evaluate products of

p

and

q

to find the pair

(p, q)

that minimizes

p \cdot q

.

p = 4

and

q = -7

, yielding the minimum product of

-28

.

If

q = 1

, the minimum product is

p \cdot 1 = -4

when

p = -4

. If

q = -7

, the product is

-7p

; to minimize this negative product, we choose the largest positive value for

p

, which is

4

, yielding

-28

. Comparing

-4

and

-28

, the absolute minimum is

-28

.

4

Compute the absolute difference

|p - q|

for the minimizing pair.

|4 - (-7)| = 11

Using the coordinates

p = 4

and

q = -7

, we find the distance between them on the number line.

Anahtar Kavram

Distance on a number line can be calculated using absolute value. Finding the minimum of a product involving signed integers requires evaluating both positive and negative cases.

Tahmini Süre:2m 0s

Soru 168Soru →

A school garden has flowers of three colors: red, yellow, and white. Red flowers make up $\frac{2}{5}$ of the garden, yellow flowers make up $\frac{1}{4}$ of the garden, and the remaining flowers are white. What percentage of the flowers in the garden are white?

33%

35%

54%

65%

67%

Cevabı ve açıklamayı göster

Cevap: 35%

Cevap

The percentage of white flowers in the garden is 35%.

To find the percentage of white flowers, we convert the fractional parts of the other flowers to percentages: red flowers make up

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

of the garden, and yellow flowers make up

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

of the garden. Adding these together, red and yellow flowers account for

40\% + 25\% = 65\%

of the garden. Subtracting this from the total garden (

100\%

) yields

100\% - 65\% = 35\%

for the white flowers.

Adım Adım Çözüm

1

Convert the fraction of red flowers to a percentage.

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

Converting all components to percentages allows for direct addition and subtraction.

2

Convert the fraction of yellow flowers to a percentage.

\frac{1}{4} = 0.25 = 25\%

This puts the yellow flowers in the same percentage unit as the red flowers.

3

Add the percentages of red and yellow flowers to find their combined share of the garden.

40\% + 25\% = 65\%

This gives the total portion of the garden occupied by non-white flowers.

4

Subtract the combined share from the total garden percentage (100%) to find the percentage of white flowers.

100\% - 65\% = 35\%

Since the entire garden represents 100%, subtracting the non-white portion yields the remaining white portion.

Anahtar Kavram

Converting fractions to percentages to solve part-to-whole word problems.

Tahmini Süre:45s

Soru 169Soru →

A school band has between $100$ and $150$ members. When the band members line up in rows of $6$ , there are $2$ members left over. When they line up in rows of $9$ , there are also $2$ members left over. When they line up in rows of $5$ , there are no members left over. How many members are in the school band?

Cevabı ve açıklamayı göster

Cevap: 110

Cevap

There are 110 members in the school band.

The number of members minus 2 must be a multiple of the least common multiple of 6 and 9, which is 18. The possible values between 100 and 150 are 110, 128, and 146. Among these, only 110 is divisible by 5, which satisfies the condition of having no members left over when grouped in rows of 5.

Adım Adım Çözüm

1

Find the least common multiple (LCM) of the row sizes 6 and 9.

The LCM of 6 and 9 is 18.

Since the remainder is the same (2) for both row sizes, the number of members minus 2 must be a common multiple of 6 and 9.

2

Identify candidate numbers between 100 and 150 that are 2 more than a multiple of 18.

The candidate numbers are 110, 128, and 146.

To satisfy the condition of having a remainder of 2 when divided by 6 and 9, the total must be of the form

18k + 2

within the given range

[100, 150]

.

3

Check which candidate is divisible by 5.

110 is divisible by 5.

Since there are no members left over when lined up in rows of 5, the total number of members must be a multiple of 5.

Anahtar Kavram

Using the least common multiple (LCM) and remainders to solve divisibility word problems.

Tahmini Süre:1m 30s

Soru 170Soru →

An agricultural drone is programmed to spray fertilizer on two adjacent fields, Field X and Field Y. The ratio of the area of Field X to the area of Field Y is $3:5$ . The drone sprays Field X at a constant rate of $4$ acres per hour, and it sprays Field Y at a constant rate of $6$ acres per hour. If it takes the drone a total of $76$ hours of continuous operation to spray both fields completely, what is the area, in acres, of Field Y?

20

48

144

240

475

Cevabı ve açıklamayı göster

Cevap: 240

Cevap

240

The area of Field Y is

240

acres. Since the ratio of the area of Field X to Field Y is

3:5

, we can define their areas as

3a

and

5a

. Using the relation

\text{Time} = \frac{\text{Area}}{\text{Rate}}

, the total time is

\frac{3a}{4} + \frac{5a}{6} = 76

. Finding a common denominator of

12

gives

\frac{9a}{12} + \frac{10a}{12} = 76

, which simplifies to

\frac{19a}{12} = 76

. Solving for

a

yields

a = 48

. The area of Field Y is then

5 \times 48 = 240

acres.

Adım Adım Çözüm

1

Define variables for the areas of the fields based on the given ratio.

Let the area of Field X be

3a

acres and the area of Field Y be

5a

acres, where

a

is a positive constant.

Since the ratio of the area of Field X to Field Y is

3:5

, expressing them in terms of a single variable

a

allows us to set up a solvable equation.

2

Express the time spent spraying each field in terms of the variable

a

.

Time for Field X =

\frac{3a}{4}

hours; Time for Field Y =

\frac{5a}{6}

hours.

Using the relation

\text{Time} = \frac{\text{Area}}{\text{Rate}}

, we substitute the respective areas and constant rates to get the time expressions.

3

Set up an equation for the total time spent spraying both fields.

\frac{3a}{4} + \frac{5a}{6} = 76

The sum of the time spent on each field must equal the total operational time of

76

hours.

4

Solve the equation for the constant

a

.

Find the common denominator of

12

:

\frac{9a}{12} + \frac{10a}{12} = 76 \implies \frac{19a}{12} = 76 \implies 19a = 912 \implies a = 48

.

Finding the least common multiple of the denominators allows us to clear fractions and isolate the variable

a

.

5

Calculate the area of Field Y.

Area of Field Y =

5 \times 48 = 240

acres.

The area of Field Y was defined as

5a

, so substituting

a = 48

gives the final area.

Anahtar Kavram

Setting up and solving equations involving ratios, rates, and proportions.

Tahmini Süre:2m 0s

Soru 171Soru →

On a certain map, a distance of $3$ inches represents an actual distance of $45$ miles. If the distance between two cities on this map is $8$ inches, what is the actual distance, in miles, between the two cities?

15

50

120

128

360

Cevabı ve açıklamayı göster

Cevap: $120$

Cevap

120

miles

Since

3

inches represents

45

miles, each inch on the map represents

\frac{45}{3} = 15

miles. To find the actual distance represented by

8

inches, multiply the number of inches by this unit rate:

8 \times 15 = 120

miles.

Adım Adım Çözüm

1

Calculate the scale of the map by finding the number of miles represented by

1

inch.

\frac{45}{3} = 15

miles per inch

This establishes the unit rate of miles per inch on the map.

2

Multiply the map distance between the two cities by the unit rate.

8 \times 15 = 120

miles

This scales the unit rate to find the actual distance for

8

inches.

Anahtar Kavram

Direct proportions and unit rates

Tahmini Süre:45s

Soru 172Soru →

A highway has exits at mile markers $-7$ and $9$ . A new rest stop is to be built at mile marker $c$ such that its distance from the exit at mile marker $-7$ is exactly $3$ times its distance from the exit at mile marker $9$ . Which of the following is a possible value of $c$ ?

-3

1

5

8

10

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value 5 is a possible mile marker for the rest stop.

The correct answer is the value 5. The distance between the rest stop at mile marker 5 and the exit at -7 is |5 - (-7)| = 12 miles. The distance between the rest stop at mile marker 5 and the exit at 9 is |5 - 9| = 4 miles. Since 12 is exactly 3 times 4, this satisfies the given condition.

Adım Adım Çözüm

1

Express the distance from the rest stop

c

to each exit using absolute value.

Distance to the exit at

-7

is

|c - (-7)| = |c + 7|

; distance to the exit at

9

is

|c - 9|

.

Absolute value represents the non-negative distance between two points on a number line.

2

Set up the algebraic equation using the given relationship.

|c + 7| = 3|c - 9|

The problem states that the distance to the exit at

-7

is 3 times the distance to the exit at

9

.

3

Solve the absolute value equation by separating it into two cases.

Case 1:

c + 7 = 3(c - 9)

or Case 2:

c + 7 = -3(c - 9)

An equation of the form

|x| = |y|

implies

x = y

or

x = -y

.

4

Solve Case 1 for

c

.

c + 7 = 3c - 27 \implies 2c = 34 \implies c = 17

Distribute the 3, subtract

c

from both sides, add 27 to both sides, and divide by 2.

5

Solve Case 2 for

c

.

c + 7 = -3c + 27 \implies 4c = 20 \implies c = 5

Distribute the -3, add

3c

to both sides, subtract 7 from both sides, and divide by 4.

6

Compare the solutions to the given options.

The two mathematically valid positions are

17

and

5

. Only

5

is listed among the options.

To identify the correct choice from the multiple-choice options.

Anahtar Kavram

Calculating distances and solving equations involving absolute values on a number line.

Alternatif Yöntem

Test the given answer choices directly by calculating the distances for each option. For the value 5, the distance to the exit at -7 is |5 - (-7)| = 12, and the distance to the exit at 9 is |5 - 9| = 4. Since 12 is 3 times 4, this value is correct.

Tahmini Süre:1m 0s

Soru 173Soru →

In a school auditorium, $\frac{3}{8}$ of the total seats are in the balcony, and the remaining seats are on the main floor. For the spring play, $80\%$ of the balcony seats were occupied, and $60\%$ of the main floor seats were occupied. If there were exactly 117 empty seats in the auditorium, what was the total number of seats in the auditorium?

Cevabı ve açıklamayı göster

Cevap: 360

Cevap

The total number of seats in the auditorium is 360.

The correct answer of 360 is found by expressing the empty seats from each section as a fraction of the total seats. Since the balcony represents

\frac{3}{8}

of the total seats and is

20\%

empty, the empty balcony seats equal

\frac{3}{40}

of the total seats. The main floor represents

\frac{5}{8}

of the total seats and is

40\%

empty, so the empty main floor seats equal

\frac{10}{40}

of the total seats. Summing these yields

\frac{13}{40}

of the total seats, which equals 117. Solving

\frac{13}{40}S = 117

gives a total of 360 seats.

Adım Adım Çözüm

1

Define the variable for the total seats and find the fraction of seats in each section.

Let the total number of seats be

S

. The balcony has

\frac{3}{8}S

seats, and the main floor has

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

seats.

Establishing fractional partitions of the total capacity is necessary to set up the equation.

2

Calculate the fraction of total seats that are empty in the balcony.

Empty balcony seats =

20\%

of

\frac{3}{8}S = 0.20 \times \frac{3}{8}S = \frac{1}{5} \times \frac{3}{8}S = \frac{3}{40}S

.

Since 80% of the balcony seats are occupied, the remaining 20% are empty.

3

Calculate the fraction of total seats that are empty on the main floor.

Empty main floor seats =

40\%

of

\frac{5}{8}S = 0.40 \times \frac{5}{8}S = \frac{2}{5} \times \frac{5}{8}S = \frac{10}{40}S

.

Since 60% of the main floor seats are occupied, the remaining 40% are empty.

4

Sum the empty seat fractions and set the sum equal to the total number of empty seats.

Total empty seats =

\frac{3}{40}S + \frac{10}{40}S = \frac{13}{40}S

. The equation is

\frac{13}{40}S = 117

.

Combining the empty seats from both sections represents the total empty seat count.

5

Solve the equation for the total number of seats

S

.

S = 117 \times \frac{40}{13} = 9 \times 40 = 360

.

Multiplying by the reciprocal of the empty seats fraction isolates the variable for the total seats.

Anahtar Kavram

Solving multi-step word problems involving fractions, decimal conversions, and percentages.

Soru 174Soru →

Three machines— $X$ , $Y$ , and $Z$ —produce widgets at constant rates. The ratio of the production rate of Machine $X$ to Machine $Y$ to Machine $Z$ is $3:4:6$ , respectively. Working together at their constant rates, the three machines can complete a standard production order in exactly $8$ hours. A new order of the same size is started with all three machines working together. After $3$ hours, Machine $Z$ breaks down and stops working, and the production rate of Machine $Y$ decreases by $25\%$ . If Machine $X$ and the slowed Machine $Y$ continue working to complete the order, what is the total number of hours required to complete this entire second order?

13\frac{5}{6}

10\frac{5}{6}

8

11\frac{1}{3}

20\frac{1}{3}

Cevabı ve açıklamayı göster

Cevap: $13\frac{5}{6}$

Cevap

13\frac{5}{6}

hours

The correct answer is

13\frac{5}{6}

hours. By translating the ratios into algebraic rates, we determine that the total order size is

104r

widgets. In the first

3

hours,

39r

widgets are produced, leaving

65r

widgets. After the breakdown and slowdown, the combined rate of Machines

X

and

Y

is

3r + 3r = 6r

widgets per hour. The remaining work requires

\frac{65r}{6r} = 10\frac{5}{6}

hours, making the total duration

3 + 10\frac{5}{6} = 13\frac{5}{6}

hours.

Adım Adım Çözüm

1

Define the rate of each machine and find the total work required for the order.

Let the rates of Machine

X

,

Y

, and

Z

be

3r

,

4r

, and

6r

widgets per hour, respectively. The combined rate is

3r + 4r + 6r = 13r

widgets per hour. The total work

W

is

13r \times 8 = 104r

widgets.

Establishing rates using the given ratios allows us to compute the size of the job in terms of rate units.

2

Calculate the work completed in the first 3 hours and find the remaining work.

Work completed in the first

3

hours is

13r \times 3 = 39r

widgets. The remaining work is

104r - 39r = 65r

widgets.

We must subtract the work done before the breakdown to find the remaining workload.

3

Determine the new rates after Machine

Z

breaks down and Machine

Y

slows down.

Machine

Z

's rate becomes

0

. Machine

Y

's new rate is

4r \times (1 - 0.25) = 3r

widgets per hour. The new combined rate for Machines

X

and

Y

is

3r + 3r = 6r

widgets per hour.

We apply the rate changes to calculate the rate at which the remaining work will be completed.

4

Calculate the time needed to complete the remaining work and add the initial time.

The remaining work takes

\frac{65r}{6r} = 10\frac{5}{6}

hours. The total time for the entire order is

3 + 10\frac{5}{6} = 13\frac{5}{6}

hours.

Dividing the remaining work by the new combined rate gives the remaining time, which we add to the initial 3 hours to get the total time.

Anahtar Kavram

Rate and ratio application to multi-stage work problems

Soru 175Soru →

Which of the following is equal to the sum of $4 \times 10^4$ and $3 \times 10^3$ , written in scientific notation?

7 \times 10^{12}

7 \times 10^7

4.3 \times 10^4

4.3 \times 10^7

1.2 \times 10^8

Cevabı ve açıklamayı göster

Cevap: $4.3 \times 10^4$

Cevap

4.3 \times 10^4

The sum of

4 \times 10^4

(which is

40,000

) and

3 \times 10^3

(which is

3,000

) is

43,000

. Written in scientific notation,

43,000

is expressed as

4.3 \times 10^4

because the decimal point is moved 4 places to the left, which corresponds to multiplying

4.3

by

10^4

.

Adım Adım Çözüm

1

Convert one of the terms so that both terms have the same power of 10.

3 \times 10^3

is rewritten as

0.3 \times 10^4

.

To add numbers in scientific notation directly, their exponents must be equal.

2

Add the coefficients while keeping the common power of 10.

(4 + 0.3) \times 10^4 = 4.3 \times 10^4

.

Adding the coefficients of terms with the same exponent yields the simplified sum in standard scientific notation.

Anahtar Kavram

Adding numbers in scientific notation requires aligning the exponents before summing the coefficients.

Alternatif Yöntem

Alternatively, you can convert both numbers to standard decimal notation first:

4 \times 10^4 = 40,000

and

3 \times 10^3 = 3,000

. Adding them gives

43,000

, which is written in standard scientific notation as

4.3 \times 10^4

.

Tahmini Süre:45s

Soru 176Soru →

A company allocates its annual marketing budget among digital, print, and television advertisements in the ratio $4:3:5$ , respectively. Due to a strategy change, the company increases its digital advertising budget by $25\%$ , decreases its print advertising budget by $40\%$ , and increases its television advertising budget by $15\%$ . If the company's total marketing budget after these changes is $\$376,500$ , what was the company's original print advertising budget?

\$30,000

\$54,000

\$90,000

\$94,125

\$120,000

Cevabı ve açıklamayı göster

Cevap: $\$90,000$

Cevap

The original print advertising budget was

\$90,000

.

The correct answer is

\$90,000

. We define the original digital, print, and television budgets as

4x

,

3x

, and

5x

, respectively. Applying the shifts yields a new digital budget of

4x \times 1.25 = 5x

, a new print budget of

3x \times 0.60 = 1.8x

, and a new television budget of

5x \times 1.15 = 5.75x

. The sum of these new budgets is

5x + 1.8x + 5.75x = 12.55x

. Setting

12.55x = 376,500

gives

x = 30,000

. The original print budget is therefore

3x = 3(30,000) = 90,000

.

Adım Adım Çözüm

1

Represent the original budget allocations using a variable

x

based on the given ratio.

Digital budget =

4x

, Print budget =

3x

, and Television budget =

5x

. The original total budget was

4x + 3x + 5x = 12x

.

Establishing algebraic terms for each part allows us to apply the percentage changes.

2

Calculate the new budget allocations after applying the percentage changes.

New Digital budget =

4x \times 1.25 = 5x

. New Print budget =

3x \times (1 - 0.40) = 1.8x

. New Television budget =

5x \times 1.15 = 5.75x

.

Applying the strategic increases and decreases determines the new individual allocations in terms of

x

.

3

Sum the new allocations and set the equation equal to the new total budget.

New Total =

5x + 1.8x + 5.75x = 12.55x

. Since the new total is

\$376,500

, we write:

12.55x = 376,500

.

This allows us to solve for the multiplier

x

.

4

Solve for

x

and calculate the original print advertising budget.

x = \frac{376,500}{12.55} = 30,000

. The original print advertising budget is

3x = 3 \times 30,000 = 90,000

.

Finding the value of

x

allows us to calculate any of the original or new budget components.

Anahtar Kavram

Solving multi-step word problems involving ratios, percentage changes, and linear equations.

Alternatif Yöntem

Alternatively, you can write the equation directly for the fraction of the budget that print represents: the original print budget is

3x

, and the new total budget is

12.55x

. The ratio of the original print budget to the new total budget is

\frac{3}{12.55}

. Multiplying this ratio by the new total budget yields

\frac{3}{12.55} \times 376,500 = 90,000

.

Tahmini Süre:2m 0s

Soru 177Soru →

If $x$ and $y$ are integers such that $|x + 3| = 4$ and $|y - 2| = 6$ , what is the maximum possible value of $|x - y|$ ?

3

7

11

13

15

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

To find the maximum possible value of

|x - y|

, we find all possible values of

x

and

y

by solving the absolute value equations. The equation

|x + 3| = 4

yields

x = 1

or

x = -7

. The equation

|y - 2| = 6

yields

y = 8

or

y = -4

. The expression

|x - y|

represents the distance between these points on a number line. The maximum distance occurs between the two points that are furthest apart, which are

y = 8

and

x = -7

. The distance between them is

|-7 - 8| = 15

.

Adım Adım Çözüm

1

Solve the absolute value equation

|x + 3| = 4

for

x

.

x + 3 = 4 \Rightarrow x = 1

or

x + 3 = -4 \Rightarrow x = -7

An absolute value equation

|u| = c

splits into two cases:

u = c

and

u = -c

.

2

Solve the absolute value equation

|y - 2| = 6

for

y

.

y - 2 = 6 \Rightarrow y = 8

or

y - 2 = -6 \Rightarrow y = -4

Similarly, split the second absolute value equation into its positive and negative cases.

3

List the possible coordinates for

x

and

y

on the number line.

x \in \{1, -7\}

and

y \in \{8, -4\}

These are the sets of values that satisfy each respective equation.

4

Find the distance

|x - y|

for all possible pairs of

(x, y)

.

For

(1, 8)

,

|1 - 8| = 7

; for

(1, -4)

,

|1 - (-4)| = 5

; for

(-7, 8)

,

|-7 - 8| = 15

; for

(-7, -4)

,

|-7 - (-4)| = 3

.

The expression

|x - y|

represents the distance between

x

and

y

on the number line. We calculate the distance for all combinations to find the maximum.

5

Identify the maximum value from the calculated distances.

The maximum value is

15

.

Comparing

7

,

5

,

15

, and

3

, the largest value is

15

.

Anahtar Kavram

Integers, Absolute Value, and Number Lines

Alternatif Yöntem

Analyze the solutions on a number line. The solutions to

|x - (-3)| = 4

are the points at a distance of

4

from

-3

, which are

-7

and

1

. The solutions to

|y - 2| = 6

are the points at a distance of

6

from

2

, which are

-4

and

8

. The maximum distance between any

x

and

y

is the distance between the leftmost point

-7

and the rightmost point

8

, which is

8 - (-7) = 15

.

Tahmini Süre:1m 0s

Soru 178Soru →

The variable $p$ represents the value $8 \times 10^5$ , and the variable $q$ represents the value $2 \times 10^{-3}$ . When the expression $\sqrt{\frac{p}{q}}$ is simplified and written in scientific notation, which of the following is the resulting value?

2 \times 10^1

2 \times 10^4

2 \times 10^8

4 \times 10^4

4 \times 10^8

Cevabı ve açıklamayı göster

Cevap: $2 \times 10^4$

Cevap

2 \times 10^4

The correct answer is obtained by first simplifying the quotient inside the radical:

\frac{8 \times 10^5}{2 \times 10^{-3}} = 4 \times 10^{5 - (-3)} = 4 \times 10^8

. Taking the square root of this value yields

\sqrt{4} \times \sqrt{10^8} = 2 \times 10^4

, which is written in standard scientific notation.

Adım Adım Çözüm

1

Substitute the values of

p

and

q

into the fraction inside the radical.

\frac{8 \times 10^5}{2 \times 10^{-3}}

To evaluate the quotient of the two numbers before applying the root.

2

Divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator.

\frac{8}{2} \times 10^{5 - (-3)} = 4 \times 10^8

Using the quotient rule for exponents:

\frac{10^a}{10^b} = 10^{a-b}

.

3

Apply the square root to both the coefficient and the power of 10.

\sqrt{4} \times \sqrt{10^8} = 2 \times 10^{8 \times \frac{1}{2}} = 2 \times 10^4

Using the properties of radicals:

\sqrt{ab} = \sqrt{a}\sqrt{b}

and

\sqrt{10^{2k}} = 10^k

.

Anahtar Kavram

Exponents, Roots, and Scientific Notation

Soru 179Soru →

On a standard number line, point $A$ is located at $-24$ and point $B$ is located at $6$ . Point $C$ is located between point $A$ and point $B$ such that the distance from $A$ to $C$ is $\frac{2}{3}$ of the distance from $C$ to $B$ . What is the coordinate of point $C$ ?

Cevabı ve açıklamayı göster

Cevap: -12

Cevap

The coordinate of point

C

is

-12

.

Because point

C

with coordinate

c

lies between

-24

and

6

, the distance from

A

to

C

is

c - (-24) = c + 24

, and the distance from

C

to

B

is

6 - c

. Setting the distance from

A

to

C

to

\frac{2}{3}

of the distance from

C

to

B

gives the equation

c + 24 = \frac{2}{3}(6 - c)

. Multiplying both sides by

3

to clear the fraction results in

3c + 72 = 12 - 2c

. Collecting like terms yields

5c = -60

, which simplifies to

c = -12

.

Adım Adım Çözüm

1

Express the distances between the points on the number line using their coordinates.

The distance from

A

to

C

is

c + 24

, and the distance from

C

to

B

is

6 - c

.

Since point

C

is positioned between points

A

and

B

, the inequality

-24 < c < 6

holds. This allows the absolute value distance expressions

|c - (-24)|

and

|6 - c|

to simplify directly to positive expressions without absolute value bars.

2

Formulate an equation based on the specified ratio of distances.

c + 24 = \frac{2}{3}(6 - c)

The problem states that the distance from

A

to

C

is

\frac{2}{3}

of the distance from

C

to

B

.

3

Solve the linear equation for the coordinate

c

.

c = -12

Multiplying both sides by

3

gives

3(c + 24) = 2(6 - c)

, which expands to

3c + 72 = 12 - 2c

. Rearranging the terms by adding

2c

to both sides and subtracting

72

from both sides results in

5c = -60

. Dividing by

5

gives

c = -12

.

Anahtar Kavram

Using absolute value properties to express distances on a number line and solving partitioning coordinate problems.

Tahmini Süre:1m 15s

Soru 180Soru →

Three cyclists start riding laps around a closed circular track at the same time from the same starting line. Cyclist A completes a lap in $\frac{2}{3}$ of a minute, Cyclist B completes a lap in $\frac{3}{4}$ of a minute, and Cyclist C completes a lap in $\frac{5}{6}$ of a minute. If they maintain these constant rates, after how many minutes will all three cyclists cross the starting line together again?

12

30

\frac{5}{2}

\frac{1}{12}

\frac{10}{13}

Cevabı ve açıklamayı göster

Cevap: $30$

Cevap

30 minutes

To find when the three cyclists meet at the starting line again, we calculate the least common multiple (LCM) of their lap times:

\frac{2}{3}

,

\frac{3}{4}

, and

\frac{5}{6}

minutes. Using the formula

\text{LCM}(\text{fractions}) = \frac{\text{LCM of numerators}}{\text{GCD of denominators}}

, the LCM of

2

,

3

, and

5

is

30

, and the GCD of

3

,

4

, and

6

is

1

. The LCM is

\frac{30}{1} = 30

minutes. At this time, each cyclist will have completed an integer number of laps (

45

,

40

, and

36

laps, respectively).

Adım Adım Çözüm

1

Identify the required mathematical concept.

We need to find the least common multiple (LCM) of the three lap times:

\frac{2}{3}

,

\frac{3}{4}

, and

\frac{5}{6}

minutes.

The next time all three cyclists cross the starting line together is the smallest positive time that is an integer multiple of each individual lap time.

2

Apply the formula for the LCM of a set of fractions.

\text{LCM}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{\text{LCM}(a, c, e)}{\text{GCD}(b, d, f)}

The LCM of fractions is determined by dividing the LCM of the numerators by the greatest common divisor (GCD) of the denominators.

3

Calculate the LCM of the numerators and the GCD of the denominators.

The LCM of

2

,

3

, and

5

is

30

. The GCD of

3

,

4

, and

6

is

1

.

For the numerators, since

2

,

3

, and

5

are prime numbers, their LCM is

2 \times 3 \times 5 = 30

. For the denominators, the only positive integer that divides

3

,

4

, and

6

is

1

.

4

Compute the final LCM value.

\frac{30}{1} = 30\text{ minutes}

Dividing the numerator LCM by the denominator GCD gives the least common multiple of the fractions.

Anahtar Kavram

Calculating the least common multiple (LCM) of a set of fractions to solve a periodic scheduling word problem.

Pre-Algebra

Cevap

Adım Adım Çözüm

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Alternatif Yöntem

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