Pre-Algebra Alıştırma Soruları — ACT — Sayfa 13

Soru 241Soru →

A certain region has a total land area of $2.5 \times 10^4$ square miles and a population of $8.0 \times 10^5$ people. What is the average population density of this region, in people per square mile?

3.2 \times 10^1

3.125 \times 10^{-2}

2.0 \times 10^{10}

3.2 \times 10^9

3.2 \times 10^{20}

Cevabı ve açıklamayı göster

Cevap: $3.2 \times 10^1$

Cevap

The population density is

3.2 \times 10^1

people per square mile.

To find the average population density, we divide the population by the land area:

\frac{8.0 \times 10^5}{2.5 \times 10^4}

. Dividing the coefficients gives

\frac{8.0}{2.5} = 3.2

. Subtracting the exponents of the powers of 10 gives

10^{5-4} = 10^1

. Combining these results yields the correct density of

3.2 \times 10^1

people per square mile.

Adım Adım Çözüm

1

Set up the ratio for population density.

\text{Density} = \frac{8.0 \times 10^5 \text{ people}}{2.5 \times 10^4 \text{ square miles}}

Population density is defined as population divided by land area.

2

Divide the numerical coefficients.

\frac{8.0}{2.5} = 3.2

To divide numbers in scientific notation, first divide their lead coefficients.

3

Divide the powers of 10 by subtracting the exponent of the denominator from the exponent of the numerator.

\frac{10^5}{10^4} = 10^{5-4} = 10^1

According to the quotient rule of exponents,

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

.

4

Combine the simplified coefficient and power of 10.

3.2 \times 10^1

This is the standard scientific notation form of the final answer.

Anahtar Kavram

Operations with numbers in scientific notation

Tahmini Süre:1m 0s

Soru 242Soru →

A student volunteer recorded the hours they spent working at a community garden and the number of plants watered each day, as shown in the table below:

Day	Hours Volunteered	Plants Watered
Monday	4	32
Tuesday	5	45
Wednesday	6	48
Thursday	4	36
Friday	8	80

What was the average (mean) number of hours volunteered per day by the student during this five-day period?

5.0

5.4

20.6

27.0

48.2

Cevabı ve açıklamayı göster

Cevap: 5.4

Cevap

5.4 hours per day

The correct answer is 5.4 hours. To find the mean, sum the daily hours volunteered (

4 + 5 + 6 + 4 + 8 = 27

) and divide by the number of days (

5

), resulting in

5.4

hours per day.

Adım Adım Çözüm

1

Calculate the total hours volunteered by summing the values in the Hours Volunteered column.

4 + 5 + 6 + 4 + 8 = 27

hours

To find the average, we first find the sum of all daily hours.

2

Divide the total hours by the number of days, which is 5.

27 \div 5 = 5.4

hours per day

The mean is defined as the sum of the data values divided by the total number of data points.

Anahtar Kavram

Calculating the arithmetic mean from a frequency table or data set.

Soru 243Soru →

An architectural firm is building a scale model of a new office building. The ratio of the height of the model to the height of the actual building is $1:50$ . The actual building will have a rectangular base with a perimeter of $270$ meters. If the ratio of the length of the base to the width of the base is $5:4$ , what is the length, in centimeters, of the base of the scale model?

1.5

120

150

300

1,500

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The length of the base of the scale model is

150

centimeters.

The correct answer represents the scaled-down length of the base converted to centimeters. First, the sum of the actual length and width is half the perimeter:

270 \div 2 = 135

meters. Since the ratio of length to width is

5:4

, we divide

135

meters into

9

equal parts of

15

meters each. The actual length is

5

parts, which equals

75

meters. Applying the scale of

1:50

gives a model length of

75 \div 50 = 1.5

meters. Finally, converting this to centimeters yields

1.5 \times 100 = 150

centimeters.

Adım Adım Çözüm

1

Calculate the sum of the actual length and width from the given perimeter.

L + W = 135

meters.

The perimeter of a rectangle is

2(L + W)

, so the sum of one length and one width is half of the perimeter:

270 \div 2 = 135

meters.

2

Use the length-to-width ratio to find the actual length of the building's base.

L = 75

meters.

The ratio

L:W = 5:4

means

L = 5x

and

W = 4x

. Substituting these into the sum gives

5x + 4x = 135 \Rightarrow 9x = 135 \Rightarrow x = 15

. Therefore, the actual length is

5 \times 15 = 75

meters.

3

Scale the actual length down to the model's scale.

Model length

= 1.5

meters.

The scale of the model is

1:50

, so the dimensions of the model are found by dividing the actual dimensions by

50

:

75 \text{ meters} \div 50 = 1.5

meters.

4

Convert the model length from meters to centimeters.

Model length

= 150

centimeters.

Since

1

meter is equal to

100

centimeters, multiplying the length in meters by

100

gives the length in centimeters:

1.5 \times 100 = 150

centimeters.

Anahtar Kavram

Solving multi-step ratio and proportion problems involving scale drawings, geometric perimeter, and unit conversions.

Alternatif Yöntem

Instead of finding the actual dimensions first, you can scale the total perimeter first: the perimeter of the model is

270 \text{ meters} \div 50 = 5.4 \text{ meters} = 540 \text{ centimeters}

. Since the ratio of length to width is

5:4

, the perimeter consists of

2(5 + 4) = 18

equal parts. The length represents

5

of these parts, so the model length is

\frac{5}{18} \times 540 = 150

centimeters.

Tahmini Süre:1m 30s

Soru 244Soru →

An agricultural irrigation system draws water from a main tank at a constant rate to supply three fields: Field X, Field Y, and Field Z. The ratio of the water volume distributed daily to Field X and Field Y is $3:4$ , and the ratio of the water volume distributed daily to Field Y and Field Z is $3:5$ . If Field Z receives $2,200$ gallons more water daily than Field X, what is the total number of gallons of water distributed daily to all three fields combined?

4,000

5,800

8,200

13,200

11,275

Cevabı ve açıklamayı göster

Cevap: $8,200$

Cevap

8,200

gallons

The correct answer is

8,200

gallons. By finding a common multiple for the Field Y term in the two ratios, we align the proportions to get a joint ratio of

9:12:20

for Field X, Field Y, and Field Z. Let the volumes be

9k

,

12k

, and

20k

. The difference between Field Z and Field X is

20k - 9k = 11k = 2,200

gallons, giving

k = 200

. The combined volume is

9k + 12k + 20k = 41k = 41 \times 200 = 8,200

gallons.

Adım Adım Çözüm

1

Find a common unit of comparison for the ratio of water distributed to all three fields.

The joint ratio is

9:12:20

.

We are given Field X to Field Y is

3:4

and Field Y to Field Z is

3:5

. To combine these ratios, we find the least common multiple of the parts representing Field Y, which is

\text{LCM}(4, 3) = 12

. Multiplying the first ratio by 3 gives

9:12

, and multiplying the second ratio by 4 gives

12:20

.

2

Set up an equation using the given difference between the water volume of Field Z and Field X.

k = 200

Let the daily volumes of Field X, Field Y, and Field Z be represented by

9k

,

12k

, and

20k

respectively. The difference between Field Z and Field X is

20k - 9k = 11k

. We are given that this difference is

2,200

gallons, so

11k = 2,200

, which simplifies to

k = 200

.

3

Calculate the combined daily volume for all three fields.

8,200

gallons

The total volume distributed to all three fields is

9k + 12k + 20k = 41k

. Substituting

k = 200

gives

41 \times 200 = 8,200

gallons.

Anahtar Kavram

Solving multi-part ratio word problems by establishing a common ratio and setting up algebraic equations.

Soru 245Soru →

A hiker starts at a base camp at an elevation of $120$ feet. She hikes up to an overlook and then down to a valley at an elevation of $-45$ feet. The elevation of the overlook is higher than both the base camp and the valley. On a vertical number line representing elevations, the distance between the overlook's coordinate and the valley's coordinate is exactly twice the distance between the overlook's coordinate and the base camp's coordinate. What is the elevation, in feet, of the overlook?

65

70

95

195

285

Cevabı ve açıklamayı göster

Cevap: 285

Cevap

285 feet

The correct answer is

285

feet. Let

x

represent the elevation of the overlook. Because the overlook is higher than both the base camp at

120

feet and the valley at

-45

feet, we know that

x > 120

. The distance between the overlook and the valley is

x - (-45) = x + 45

, and the distance between the overlook and the base camp is

x - 120

. Setting up the relationship where the distance to the valley is twice the distance to the base camp gives the equation

x + 45 = 2(x - 120)

. Solving this equation yields

x + 45 = 2x - 240

, which simplifies to

x = 285

. This value is indeed higher than both elevations, and we can verify that the distance to the valley (

285 - (-45) = 330

) is twice the distance to the base camp (

285 - 120 = 165

).

Adım Adım Çözüm

1

Define the variable and write the expressions for the distances on a number line.

Let

x

represent the elevation of the overlook in feet. The distance between the overlook and the base camp (

120

feet) is

|x - 120|

. The distance between the overlook and the valley (

-45

feet) is

|x - (-45)| = |x + 45|

.

Distance between two points

a

and

b

on a number line is defined by the absolute value of their difference,

|a - b|

.

2

Set up the equation based on the given ratio.

|x + 45| = 2|x - 120|

The problem states that the distance to the valley is exactly twice the distance to the base camp.

3

Apply the condition that the overlook is higher than both locations to simplify the absolute value expressions.

Since

x > 120

and

x > -45

, both terms inside the absolute values are positive. Therefore,

|x + 45| = x + 45

and

|x - 120| = x - 120

. The equation simplifies to:

x + 45 = 2(x - 120)

.

For any expression

y > 0

,

|y| = y

. Establishing the location of

x

relative to

120

and

-45

allows us to remove the absolute value bars.

4

Solve the simplified linear equation for

x

.

x + 45 = 2x - 240 \Rightarrow x = 285

Distribute the

2

on the right side and isolate

x

by subtracting

x

and adding

240

to both sides.

Anahtar Kavram

Representing distances between points on a number line using absolute value equations.

Alternatif Yöntem

Instead of setting up equations, you can test the options starting from the constraint that the elevation must be greater than

120

feet. This eliminates

65

,

70

, and

95

. Test the remaining options: for

195

, the distance to

120

is

75

and the distance to

-45

is

240

(not twice

75

). For

285

, the distance to

120

is

165

and the distance to

-45

is

330

, which is exactly twice

165

.

Tahmini Süre:1m 30s

Soru 246Soru →

An online retailer determines the sale price of a promotional item by reducing the original price, $P$ dollars, by a discount of $D$ dollars. The discount is defined as $D = \frac{1}{6} P$ . If the sale price of the item is $125$ dollars, which of the following equations can be solved to find the original price of the item, in dollars?

\frac{1}{6} P = 125

P - \frac{1}{6} = 125

\frac{5}{6} P = 125

P + \frac{1}{6} P = 125

\frac{6}{5} P = 125

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{6} P = 125$

Cevap

The equation stating that five-sixths of the original price equals one hundred twenty-five is correct.

The correct equation is found by expressing the sale price as the original price minus the discount. The discount is one-sixth of the original price, which is represented by the expression containing five-sixths of the original price. Setting this expression equal to the given sale price of one hundred twenty-five dollars results in the correct equation.

Adım Adım Çözüm

1

Write the relationship for the sale price based on the scenario.

S = P - D

The sale price (

S

) is the original price (

P

) minus the discount amount (

D

).

2

Substitute the given expression for the discount (

D = \frac{1}{6} P

) into the sale price equation.

S = P - \frac{1}{6} P

This expresses the sale price in terms of the single variable

P

representing the original price.

3

Simplify the algebraic expression by combining the like terms.

S = \frac{5}{6} P

Combining

1P

and

-\frac{1}{6} P

gives

\frac{5}{6} P

because

1 - \frac{1}{6} = \frac{5}{6}

.

4

Substitute the known sale price of

125

dollars into the simplified equation.

\frac{5}{6} P = 125

This sets up the final one-step equation needed to find the original price.

Anahtar Kavram

Translating verbal descriptions of discounts into simplified one-step linear equations.

Tahmini Süre:1m 30s

Soru 247Soru →

A manufacturer of solar panels inspects their production in three stages. In the first stage, $\frac{1}{12}$ of the total panels produced are identified as defective and recycled. In the second stage, $20\%$ of the remaining non-defective panels are found to have cosmetic flaws and are sold at a discount. In the third stage, $\frac{1}{5}$ of the panels that passed the first two stages without any defects or flaws are selected for quality testing. If $1,760$ panels passed the first two stages but were NOT selected for quality testing, what was the total number of panels originally produced?

Cevabı ve açıklamayı göster

Cevap: 3000

Cevap

The total number of panels originally produced was

3,000

.

The correct answer of

3,000

is determined by calculating the cumulative fraction of panels that are not defective, do not have cosmetic flaws, and are not selected for testing. In Stage 1, subtracting

\frac{1}{12}

leaves

\frac{11}{12}

of the original panels. In Stage 2, subtracting

20\%

of those leaves

80\%

(or

\frac{4}{5}

), resulting in

\frac{11}{15}

of the original panels. In Stage 3, subtracting

\frac{1}{5}

of these leaves

\frac{4}{5}

, resulting in

\frac{44}{75}

of the original panels. Setting

\frac{44}{75}

of the original panels equal to

1,760

and solving for the total original panels yields

3,000

.

Adım Adım Çözüm

1

Calculate the fraction of panels remaining after the first stage of inspection.

\frac{11}{12}

of the original panels

Since

\frac{1}{12}

of the total panels are defective and recycled, the remaining portion is

1 - \frac{1}{12} = \frac{11}{12}

.

2

Calculate the fraction of the original panels that pass the second stage without defects or flaws.

\frac{11}{15}

of the original panels

Of the remaining

\frac{11}{12}

of the panels,

20\%

have cosmetic flaws, meaning

100\% - 20\% = 80\%

(or

\frac{4}{5}

) do not have flaws. Multiplying these fractions gives

\frac{4}{5} \times \frac{11}{12} = \frac{11}{15}

.

3

Calculate the fraction of original panels that pass the first two stages but are not selected for testing.

\frac{44}{75}

of the original panels

Since

\frac{1}{5}

of the flawless panels are selected for testing, the remaining

1 - \frac{1}{5} = \frac{4}{5}

are not selected. Multiplying this by the fraction from Step 2 gives

\frac{4}{5} \times \frac{11}{15} = \frac{44}{75}

.

4

Solve for the original number of panels,

P

, using the given value of

1,760

.

P = 3,000

panels

Set up the equation

\frac{44}{75} P = 1,760

. Solving for

P

gives

P = 1,760 \times \frac{75}{44} = 40 \times 75 = 3,000

.

Anahtar Kavram

Applying successive fractions and percentages to resolve a multi-stage word problem by working backward to the initial value.

Soru 248Soru →

If $k$ and $m$ are integers such that $|k - 3| = 8$ and $|m + 2| = 5$ , where $k < 0$ and $m < 0$ , what is the distance on the number line between $k$ and $m$ ?

2

4

8

12

18

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The distance on the number line between

k

and

m

is

2

.

To find

k

, solve

|k - 3| = 8

, which gives

k - 3 = 8 \implies k = 11

or

k - 3 = -8 \implies k = -5

. Since

k < 0

,

k = -5

. To find

m

, solve

|m + 2| = 5

, which gives

m + 2 = 5 \implies m = 3

or

m + 2 = -5 \implies m = -7

. Since

m < 0

,

m = -7

. The distance on the number line between

-5

and

-7

is

|-5 - (-7)| = 2

.

Adım Adım Çözüm

1

Solve the absolute value equation for

k

using the given constraint

k < 0

.

k - 3 = 8 \implies k = 11

or

k - 3 = -8 \implies k = -5

. Since

k < 0

,

k = -5

.

An absolute value equation

|X| = c

splits into

X = c

and

X = -c

.

2

Solve the absolute value equation for

m

using the given constraint

m < 0

.

m + 2 = 5 \implies m = 3

or

m + 2 = -5 \implies m = -7

. Since

m < 0

,

m = -7

.

An absolute value equation

|X| = c

splits into

X = c

and

X = -c

.

3

Calculate the distance between

k = -5

and

m = -7

on the number line.

\text{Distance} = |k - m| = |-5 - (-7)| = |-5 + 7| = |2| = 2.

The distance between two points on a number line is given by the absolute value of their difference.

Anahtar Kavram

Evaluating absolute value equations and finding distance between two integers on a number line.

Tahmini Süre:1m 15s

Soru 249Soru →

A public library is organizing its catalog of physical books. Currently, $\frac{3}{8}$ of the books in the collection are categorized as fiction. Of the remaining books, $40\%$ are non-fiction, while the rest are divided between children's literature and reference volumes in a ratio of $3:2$ , respectively. If there are exactly $450$ reference volumes in the library, what is the total number of physical books in the collection?

1,350

1,800

1,875

3,000

5,000

Cevabı ve açıklamayı göster

Cevap: $3,000$

Cevap

The total number of physical books in the library collection is

3,000

.

To find the total number of books, first calculate the remaining fraction after fiction books are removed:

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

. Next, account for non-fiction taking up

40\%

of this remainder, leaving

60\%

(or

\frac{3}{5}

) of

\frac{5}{8}

for children's and reference books, which equals

\frac{3}{8}

of the entire collection. Using the

3:2

ratio, reference books represent

\frac{2}{5}

of this

\frac{3}{8}

portion, which is

\frac{3}{20}

of the total collection. Setting

\frac{3}{20}

of the total equal to

450

gives a total collection of

3,000

books.

Adım Adım Çözüm

1

Determine the remaining fraction of books after setting aside fiction books.

Since

\frac{3}{8}

of the books are fiction, the remaining fraction is

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

.

The total collection equals

1

(or

100\%

). Subtracting the fiction fraction leaves the remaining portion.

2

Calculate the fraction of books that are either children's literature or reference volumes.

Since

40\%

of the remaining books are non-fiction, the remaining

60\%

(

0.60

or

\frac{3}{5}

) of the

\frac{5}{8}

portion are children's or reference books:

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

.

Finding

60\%

of the remaining

\frac{5}{8}

identifies the portion shared between children's and reference books.

3

Determine the fraction of total books represented by reference volumes using the given ratio.

With a children's to reference ratio of

3:2

, reference books represent

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

of this subgroup. Thus, reference books account for

\frac{2}{5} \times \frac{3}{8} = \frac{6}{40} = \frac{3}{20}

of the total collection.

A ratio of

3:2

divides the subgroup into

5

equal parts, of which reference books make up

2

parts.

4

Solve for the total collection size using the known number of reference volumes.

Set up the equation

\frac{3}{20} \times T = 450

, where

T

is the total number of books. Solving yields

T = 450 \times \frac{20}{3} = 150 \times 20 = 3,000

.

Dividing the given quantity by its corresponding fraction gives the total population size.

Anahtar Kavram

Fractions, Decimals, and Percentages

Tahmini Süre:2m 0s

Soru 250Soru →

If $a$ and $b$ are nonzero integers such that $|a| < |b|$ and $a \cdot b < 0$ , which of the following expressions must have the greatest value?

|a - b|

|a + b|

a - b

|a| - |b|

|b| - |a|

Cevabı ve açıklamayı göster

Cevap: $|a - b|$

Cevap

The expression

|a - b|

must have the greatest value because

a

and

b

have opposite signs, making

|a - b| = |a| + |b|

.

Because

a

and

b

have opposite signs (

a \cdot b < 0

), the distance between them on the number line is the sum of their individual distances from zero,

|a| + |b|

. The expression

|a - b|

measures this exact distance, which is guaranteed to be greater than any net difference or signed value.

Adım Adım Çözüm

1

Analyze the given conditions

Since

a \cdot b < 0

,

a

and

b

have opposite signs (one positive, one negative). Since

|a| < |b|

, the absolute value of

b

is strictly greater than the absolute value of

a

.

Determining the signs and relative magnitudes of

a

and

b

is necessary to evaluate absolute value expressions.

2

Evaluate

|a - b|

in terms of magnitudes

When two numbers have opposite signs, subtracting them adds their magnitudes:

|a - b| = |a| + |b|

.

If

a > 0

and

b < 0

,

|a - b| = a - (b) = |a| + |b|

. If

a < 0

and

b > 0

,

|a - b| = |- |a| - |b|| = |a| + |b|

.

3

Compare

|a - b|

to the other expressions

|a - b| = |a| + |b|

is strictly greater than

|a + b| = |b| - |a|

,

|b| - |a|

,

|a| - |b|

, and

a - b

.

Since both

a

and

b

are non-zero,

|a| + |b| > |b| - |a| > 0 > |a| - |b|

.

Anahtar Kavram

Absolute value distance and sign properties on a number line

Soru 251Soru →

On a standard number line, point $R$ has coordinate $-18$ and point $T$ has coordinate $22$ . Point $S$ lies on the line segment $RT$ such that the distance from $R$ to $S$ is $\frac{3}{5}$ of the distance from $R$ to $T$ . What is the coordinate of point $S$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The distance between

R(-18)

and

T(22)

is calculated using absolute value:

|22 - (-18)| = 40

. Taking

\frac{3}{5}

of 40 yields 24 units. Adding 24 to the starting coordinate

-18

gives the coordinate of point

S

as 6.

Adım Adım Çözüm

1

Calculate the total distance between points

R

and

T

Distance

= |22 - (-18)| = |40| = 40

units

The distance between two points on a number line is given by the absolute difference of their coordinates.

2

Determine the distance from

R

to

S

Distance

= \frac{3}{5} \times 40 = 24

units

Point

S

is located at a distance equal to

\frac{3}{5}

of the total segment length starting from

R

.

3

Calculate the coordinate of point

S

Coordinate of

S = -18 + 24 = 6

Since

S

lies between

R(-18)

and

T(22)

, moving from

R

toward

T

corresponds to increasing the coordinate value by 24.

Anahtar Kavram

Calculating distance between signed integers using absolute value and finding a point dividing a number line segment in a given ratio

Soru 252Soru →

A community solar farm distributes electrical energy to three types of facilities: residential homes, small businesses, and municipal buildings. The ratio of energy distributed to residential homes to energy distributed to small businesses is $5:3$ . The ratio of energy distributed to small businesses to energy distributed to municipal buildings is $4:1$ . If the municipal buildings receive a total of $45$ megawatt-hours (MWh) of energy, what is the total amount of energy, in megawatt-hours, distributed by the solar farm?

255

300

450

480

525

Cevabı ve açıklamayı göster

Cevap: 525

Cevap

The total energy distributed by the solar farm is 525 megawatt-hours.

To solve this problem, express the two ratios with a unified scale. Residential to Small Businesses is

5:3

and Small Businesses to Municipal is

4:1

. Multiplying the first ratio by

4

gives

20:12

and the second ratio by

3

gives

12:3

. This yields a three-part ratio of

20:12:3

for Residential : Small Businesses : Municipal. Since Municipal buildings account for

3

units out of the total

35

units (

20 + 12 + 3 = 35

) and receive

45

MWh, each ratio unit corresponds to

\frac{45}{3} = 15

MWh. Multiplying the total

35

units by

15

MWh gives

525

MWh.

Adım Adım Çözüm

1

Express both ratios using a common term for small businesses.

Residential to Small Businesses

= 5:3 = 20:12

. Small Businesses to Municipal

= 4:1 = 12:3

. Combining these gives a ratio of Residential to Small Businesses to Municipal of

20:12:3

.

To combine two ratios sharing a common quantity, scale the ratios so the common quantity has the same numerical value in both.

2

Determine the value of one ratio unit.

Municipal buildings equal

3

ratio units, which corresponds to

45

MWh. Therefore,

1

ratio unit

= \frac{45}{3} = 15

MWh.

Knowing the actual quantity associated with one component allows us to find the constant multiplier for the ratio.

3

Calculate the total energy distributed.

Total ratio units

= 20 + 12 + 3 = 35

units. Total energy

= 35 \times 15 = 525

MWh.

Multiply the sum of all ratio parts by the value of a single ratio unit to find the grand total.

Anahtar Kavram

Combining Compound Ratios and Solving Proportions

Soru 253Soru →

A retail clothing store applies successive markdowns to a coat during a seasonal clearance sale. In the first week, the original price of the coat is discounted by $30\%$ . In the second week, the price is reduced by an additional $\frac{1}{5}$ of the first-week sale price. Finally, a customer uses a promotional code at checkout to receive an extra $15\%$ off the second-week sale price. If the customer pays a final price of $\$119.00$ before tax, what was the original price of the coat, in dollars?

Cevabı ve açıklamayı göster

Cevap: 250

Cevap

The original price of the coat was

250

dollars.

To find the original price of the coat, express each successive discount as a multiplier representing the portion of the remaining price. The first discount of

30\%

leaves

70\%

(

0.70

). The second discount of

\frac{1}{5}

(

20\%

) leaves

80\%

(

0.80

) of the first sale price. The final discount of

15\%

leaves

85\%

(

0.85

) of the second sale price. Combining these yields an overall price multiplier of

0.70 \times 0.80 \times 0.85 = 0.476

. Dividing the final purchase price of

\$119.00

by

0.476

gives the original price of

\$250.00

.

Adım Adım Çözüm

1

Calculate the price multiplier for the first week discount.

Multiplier 1 =

0.70

A

30\%

discount leaves

100\% - 30\% = 70\%

of the original price.

2

Convert the second week fraction discount to a decimal multiplier.

Multiplier 2 =

0.80

A discount of

\frac{1}{5}

is equal to

20\%

, leaving

1 - 0.20 = 0.80

of the week-1 price.

3

Calculate the price multiplier for the promotional code.

Multiplier 3 =

0.85

A

15\%

discount leaves

100\% - 15\% = 85\%

(

0.85

) of the week-2 price.

4

Find the combined price multiplier by taking the product of all three individual multipliers.

Combined Multiplier =

0.476

Successive discounts compound multiplicatively:

0.70 \times 0.80 \times 0.85 = 0.476

.

5

Solve for the original price

P

using the equation

0.476 P = 119

.

P = 250

Dividing the final price

\$119.00

by

0.476

yields

250

.

Anahtar Kavram

Compounding successive percentage and fraction discounts and solving for the original value.

Tahmini Süre:2m 0s

Soru 254Soru →

A high-speed laser pulse travels at a constant speed of $3.0 \times 10^8$ meters per second. If the pulse takes $4.5 \times 10^{-7}$ seconds to travel the entire length of a specialized optical fiber, what is the length of the optical fiber, in meters?

Cevabı ve açıklamayı göster

Cevap: 135

Cevap

The length of the optical fiber is 135 meters.

Multiplying the speed

3.0 \times 10^8

meters per second by the time

4.5 \times 10^{-7}

seconds yields

(3.0 \times 4.5) \times 10^{8 + (-7)} = 13.5 \times 10^1 = 135

meters.

Adım Adım Çözüm

1

Set up the distance equation

d = (3.0 \times 10^8) \times (4.5 \times 10^{-7})

Distance equals speed multiplied by time.

2

Group coefficients and powers of 10

d = (3.0 \times 4.5) \times (10^8 \times 10^{-7})

The commutative and associative properties of multiplication allow grouping coefficients and base-10 exponential terms together.

3

Simplify the coefficients and add exponents

d = 13.5 \times 10^1 = 135

Multiplying the coefficients yields 13.5, and adding the exponents gives

8 + (-7) = 1

. Evaluating

13.5 \times 10^1

results in 135.

Anahtar Kavram

Multiplying numbers expressed in scientific notation using exponent rules.

Soru 255Soru →

On a standard number line, point $A$ has coordinate $-28$ and point $B$ has coordinate $12$ . Point $C$ is the midpoint of segment $AB$ . Point $D$ is positioned on the line such that point $B$ is the midpoint of segment $CD$ . What is the distance between point $C$ and point $D$ on the number line?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The distance between point

C

and point

D

on the number line is 40.

The midpoint

C

of segment

AB

is found by averaging the coordinates:

\frac{-28 + 12}{2} = -8

. Because point

B(12)

is the midpoint of segment

CD

, set up the equation

\frac{-8 + D}{2} = 12

, yielding

D = 32

. The distance between point

C(-8)

and point

D(32)

is the absolute value of their difference:

|32 - (-8)| = 40

.

Adım Adım Çözüm

1

Calculate the coordinate of point

C

, which is the midpoint of segment

AB

.

The coordinate of point

C

is

-8

.

The midpoint of two points on a number line is given by their average:

\frac{-28 + 12}{2} = -8

.

2

Determine the coordinate of point

D

using the midpoint relationship for segment

CD

.

The coordinate of point

D

is

32

.

Since point

B(12)

is the midpoint of segment

CD

,

\frac{C + D}{2} = 12 \implies \frac{-8 + D}{2} = 12

, which solves to

D = 32

.

3

Find the distance between point

C

and point

D

.

The distance is

40

.

Distance on a number line is the absolute value of the difference between coordinates:

|32 - (-8)| = |40| = 40

.

Anahtar Kavram

Midpoint and Absolute Value Distance on a Number Line

Alternatif Yöntem

Alternatively, note that the distance of segment

AB

is

|12 - (-28)| = 40

. Since

C

is the midpoint of

AB

, the length of segment

CB

is

\frac{40}{2} = 20

. Since

B

is the midpoint of segment

CD

, the length of segment

BD

must equal the length of segment

CB

, which is

20

. Therefore, the total distance from

C

to

D

is

CB + BD = 20 + 20 = 40

.

Tahmini Süre:1m 15s

Soru 256Soru →

Let $a$ and $b$ be integers. On a standard number line, $a = -14$ , and the distance between $a$ and $b$ is $23$ . If $b > 0$ , what is the value of $|2b - |a||$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

|2b - |a||

is 4.

The distance between

a = -14

and

b

on the number line is

|b - (-14)| = |b + 14| = 23

. Solving for a positive value of

b

gives

b + 14 = 23

, so

b = 9

. The absolute value of

a

is

|-14| = 14

. Substituting these values into

|2b - |a||

yields

|2(9) - 14| = |18 - 14| = 4

.

Adım Adım Çözüm

1

Determine the value of integer

b

using the distance relationship on the number line.

The distance formula

|b - (-14)| = 23

simplifies to

|b + 14| = 23

. Since

b > 0

,

b = 9

.

The distance between two points

x

and

y

on a number line is given by

|x - y|

.

2

Calculate the absolute value of

a

.

|a| = |-14| = 14

.

The absolute value of a negative number is its positive magnitude.

3

Substitute

b = 9

and

|a| = 14

into the expression

|2b - |a||

.

|2(9) - 14| = |18 - 14| = 4

.

Perform operations inside the outer absolute value first, then take the absolute value of the result.

Anahtar Kavram

Absolute Value as Distance on a Number Line

Tahmini Süre:1m 0s

Soru 257Soru →

A coffee artisan roasts a $120$ -pound batch of green coffee beans. During the roasting process, the beans lose $15\%$ of their total weight due to moisture evaporation. Next, $\frac{1}{8}$ of the roasted weight is removed during quality control sorting. Finally, the remaining high-quality roasted beans are packaged into small bags that each hold $0.75$ pounds of coffee. How many full bags of roasted coffee can be produced from this batch?

116

119

122

136

161

Cevabı ve açıklamayı göster

Cevap: $119$

Cevap

The total number of full bags of roasted coffee that can be produced is

119

.

To find the number of bags, calculate the remaining weight step-by-step. First,

15\%

of

120

pounds is

18

pounds, leaving

102

pounds of roasted coffee. Next,

\frac{1}{8}

of

102

pounds is

12.75

pounds, leaving

89.25

pounds of high-quality coffee. Dividing

89.25

by

0.75

pounds per bag yields exactly

119

full bags.

Adım Adım Çözüm

1

Calculate the weight lost during roasting

15\%

of

120\text{ lbs} = 0.15 \times 120 = 18\text{ lbs}

The beans lose

15\%

of their initial weight due to evaporation.

2

Determine the remaining weight of roasted beans

120\text{ lbs} - 18\text{ lbs} = 102\text{ lbs}

Subtract the weight lost during roasting from the starting weight.

3

Calculate the weight removed during quality control

\frac{1}{8} \times 102\text{ lbs} = 12.75\text{ lbs}

Quality control removes

\frac{1}{8}

of the roasted weight.

4

Determine the final usable weight of coffee beans

102\text{ lbs} - 12.75\text{ lbs} = 89.25\text{ lbs}

Subtract the rejected beans from the roasted weight.

5

Calculate the number of

0.75

-pound bags produced

89.25 \div 0.75 = 119\text{ bags}

Divide the total usable weight by the capacity of a single bag.

Anahtar Kavram

Sequential application of percentages and fractions to remaining quantities

Tahmini Süre:2m 0s

Soru 258Soru →

Star X emits $4.8 \times 10^{22}$ Joules of energy per second, which is 16 times the energy emitted per second by Star Y. What is the energy, in Joules, emitted per second by Star Y, expressed in scientific notation?

7.68 \times 10^{23}

3.0 \times 10^{23}

3.0 \times 10^{22}

3.0 \times 10^{21}

3.0 \times 10^{20}

Cevabı ve açıklamayı göster

Cevap: $3.0 \times 10^{21}$

Cevap

3.0 \times 10^{21}

The energy emitted by Star Y is found by dividing Star X's energy output (

4.8 \times 10^{22}

) by 16. Dividing

4.8

by 16 yields

0.3

. To put

0.3 \times 10^{22}

in standard scientific notation, move the decimal point one place to the right, which reduces the exponent from 22 to 21, resulting in

3.0 \times 10^{21}

.

Adım Adım Çözüm

1

Translate the relationship into an equation.

\text{Energy}_X = 16 \times \text{Energy}_Y

, so

\text{Energy}_Y = \frac{\text{Energy}_X}{16} = \frac{4.8 \times 10^{22}}{16}

.

Since Star X emits 16 times as much energy as Star Y, dividing Star X's energy output by 16 gives Star Y's energy output.

2

Divide the numerical coefficient by 16.

\frac{4.8}{16} \times 10^{22} = 0.3 \times 10^{22}

Perform the division on the non-exponent part of the scientific notation expression.

3

Convert

0.3 \times 10^{22}

into standard scientific notation.

0.3 \times 10^{22} = (3.0 \times 10^{-1}) \times 10^{22} = 3.0 \times 10^{21}

Standard scientific notation requires the coefficient

a

to satisfy

1 \le |a| < 10

. Moving the decimal point one place to the right decreases the exponent of 10 by 1.

Anahtar Kavram

Division of Numbers in Scientific Notation

Soru 259Soru →

A specialty beverage company creates a signature tea blend by combining Black Tea, Green Tea, and White Tea leaves in the ratio $5 : 3 : 2$ by weight. A master blender prepares a batch of this signature blend that contains $18$ pounds more Black Tea leaves than Green Tea leaves. What is the total weight, in pounds, of this batch of signature tea blend?

36

45

60

72

90

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

The total weight of the batch of signature tea blend is 90 pounds.

Let

x

represent the weight in pounds of one ratio part. The weights of Black Tea, Green Tea, and White Tea are

5x

,

3x

, and

2x

pounds, respectively. Since the batch has

18

pounds more Black Tea than Green Tea, we set up the equation

5x - 3x = 18

, which simplifies to

2x = 18

, giving

x = 9

. The total ratio sum for the blend is

5 + 3 + 2 = 10

parts. Thus, the total weight of the batch is

10 \times 9 = 90

pounds.

Adım Adım Çözüm

1

Define the ratio quantities using a common variable x.

Black Tea weight =

5x

, Green Tea weight =

3x

, White Tea weight =

2x

.

The given ratio of Black Tea to Green Tea to White Tea is

5 : 3 : 2

.

2

Set up an equation based on the given difference between Black Tea and Green Tea.

5x - 3x = 18 \implies 2x = 18 \implies x = 9

.

The batch contains 18 pounds more Black Tea leaves than Green Tea leaves.

3

Calculate the total ratio parts and determine the total weight.

Total parts =

5 + 3 + 2 = 10

. Total weight =

10x = 10 \times 9 = 90

pounds.

The total weight is the sum of all three tea leaf components.

Anahtar Kavram

Solving multi-part ratio word problems using a scaling multiplier based on part-to-part differences.

Tahmini Süre:1m 0s

Soru 260Soru →

A digital commercial printing press operates at a constant rate of $r$ pages per minute. During a scheduled production session lasting $t$ minutes, the machine was paused for $12$ minutes due to paper replenishment. If the printing press successfully printed $p$ pages during the operational portion of the session, which of the following equations correctly expresses the relationship between $r$ , $p$ , and $t$ ?

r = \frac{p}{t} - 12

r = \frac{p}{t - 12}

r = \frac{p}{t + 12}

r = \frac{p - 12}{t}

r = \frac{12p}{t}

Cevabı ve açıklamayı göster

Cevap: $r = \frac{p}{t - 12}$

Cevap

The correct equation is

r = \frac{p}{t - 12}

.

Since the press was paused for 12 minutes during a session of

t

minutes, the total time spent actually printing was

(t - 12)

minutes. The printing rate

r

is defined as pages printed divided by active printing time, giving

r = \frac{p}{t - 12}

.

Adım Adım Çözüm

1

Identify the active printing time in terms of

t

.

Active operating time =

(t - 12)

minutes.

The machine was paused for 12 minutes out of the total session duration of

t

minutes.

2

Use the basic rate relationship Rate = Quantity / Time.

r = \frac{p}{t - 12}

.

Dividing the total pages printed

p

by the active printing duration

(t - 12)

yields the rate

r

in pages per minute.

Anahtar Kavram

Formulating one-step rate and time algebraic relationships with adjusted variables.

Pre-Algebra

Cevap

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Alternatif Yöntem

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Alternatif Yöntem

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