Pre-Algebra Alıştırma Soruları — ACT — Sayfa 11

Soru 201Soru →

On a number line, point $M$ has coordinate $-14$ and point $N$ has coordinate $10$ . Point $P$ is located on the number line such that the distance between $M$ and $P$ is three times the distance between $N$ and $P$ . If the coordinate of $P$ is positive, what is the sum of all possible coordinates of $P$ ?

4

8

22

26

34

Cevabı ve açıklamayı göster

Cevap: $26$

Cevap

The sum of all possible coordinates of

P

is

26

.

The distance between any two points

x

and

y

on a number line is given by

|x - y|

. Thus, the distance from point

P(p)

to

M(-14)

is

|p - (-14)| = |p + 14|

, and the distance from

P(p)

to

N(10)

is

|p - 10|

. Since the distance to

M

is three times the distance to

N

, we write

|p + 14| = 3|p - 10|

. To solve this absolute value equation, we check two cases. In the first case, we have

p + 14 = 3(p - 10)

, which simplifies to

2p = 44

, or

p = 22

. In the second case, we have

p + 14 = -3(p - 10)

, which simplifies to

4p = 16

, or

p = 4

. Both coordinates are positive, satisfying the condition given in the problem. The sum of these possible coordinates is

22 + 4 = 26

.

Adım Adım Çözüm

1

Set up the algebraic representation of the distances between the points on the number line using absolute value.

The distance between point

M(-14)

and point

P(p)

is

|p - (-14)| = |p + 14|

. The distance between point

N(10)

and point

P(p)

is

|p - 10|

.

The distance between two points

a

and

b

on a standard number line is always expressed as the absolute value of their difference,

|a - b|

.

2

Formulate the equation representing the relationship between the two distances.

|p + 14| = 3|p - 10|

The problem states that the distance from

M

to

P

is three times the distance from

N

to

P

.

3

Solve the absolute value equation by analyzing both positive and negative cases.

Case 1:

p + 14 = 3(p - 10) \implies p + 14 = 3p - 30 \implies 2p = 44 \implies p = 22

.
Case 2:

p + 14 = -3(p - 10) \implies p + 14 = -3p + 30 \implies 4p = 16 \implies p = 4

.

An equation of the form

|A| = |B|

is solved by evaluating the two distinct possibilities:

A = B

and

A = -B

.

4

Check if both solutions satisfy the condition of being positive, and sum them.

Both

22

and

4

are positive, so they are both valid coordinates. Their sum is

22 + 4 = 26

.

The question specifies that the coordinate of

P

must be positive, and asks for the sum of all such coordinates.

Anahtar Kavram

Representing distances on a number line using absolute value and solving absolute value equations.

Tahmini Süre:1m 30s

Soru 202Soru →

In astrophysics, the energy intensities of two different cosmic signals are modeled. The primary signal has an intensity of $I_1 = \sqrt[3]{x \cdot y^2}$ watts per square meter, where $x = 4.0 \times 10^4$ and $y = 5.0 \times 10^3$ . The secondary signal has an intensity of $I_2 = \frac{\sqrt{z^3 \cdot w^{-1}}}{2.0 \times 10^{-2}}$ watts per square meter, where $z = 1.6 \times 10^{-3}$ and $w = 2.5 \times 10^{-7}$ . What is the ratio of the primary signal intensity to the secondary signal intensity, expressed in scientific notation?

6.25 \times 10^2

6.4 \times 10^{-4}

1.5625 \times 10^2

1.5625 \times 10^3

1.5625 \times 10^7

Cevabı ve açıklamayı göster

Cevap: $1.5625 \times 10^3$

Cevap

The correct ratio of the primary signal intensity to the secondary signal intensity is

1.5625 \times 10^3

.

To find the correct ratio of the primary signal intensity to the secondary signal intensity, first compute the value of both intensities. The primary signal intensity is

I_1 = \sqrt[3]{x \cdot y^2} = \sqrt[3]{4.0 \times 10^4 \cdot (5.0 \times 10^3)^2} = \sqrt[3]{1.0 \times 10^{12}} = 1.0 \times 10^4 = 10,000

. The secondary signal intensity is evaluated by simplifying the numerator term:

z^3 \cdot w^{-1} = (1.6 \times 10^{-3})^3 \cdot (2.5 \times 10^{-7})^{-1} = 4.096 \times 10^{-9} \cdot 4.0 \times 10^6 = 1.6384 \times 10^{-2}

. Taking the square root gives

1.28 \times 10^{-1}

. Dividing this by the denominator yields

I_2 = \frac{1.28 \times 10^{-1}}{2.0 \times 10^{-2}} = 6.4

. Finally, the ratio of the primary intensity to the secondary intensity is

\frac{10000}{6.4} = 1562.5

, which is expressed in standard scientific notation as

1.5625 \times 10^3

.

Adım Adım Çözüm

1

Calculate the squared value of the variable y to evaluate the primary signal intensity.

y^2 = (5.0 \times 10^3)^2 = 5.0^2 \times (10^3)^2 = 25 \times 10^6 = 2.5 \times 10^7

Squaring a term in scientific notation requires squaring the coefficient and multiplying the exponent of 10 by 2.

2

Multiply the variable x by the calculated value of y^2 and take the cube root to find the primary intensity.

x \cdot y^2 = (4.0 \times 10^4) \times (2.5 \times 10^7) = 10 \times 10^{11} = 1.0 \times 10^{12}

; therefore,

I_1 = \sqrt[3]{1.0 \times 10^{12}} = 1.0 \times 10^4 = 10,000

To multiply numbers in scientific notation, multiply their coefficients and add their exponents. To find the cube root, raise both parts to the 1/3 power.

3

Evaluate the expressions for z^3 and w^{-1} in the numerator of the secondary signal intensity.

z^3 = (1.6 \times 10^{-3})^3 = 4.096 \times 10^{-9}

and

w^{-1} = (2.5 \times 10^{-7})^{-1} = 0.4 \times 10^7 = 4.0 \times 10^6

To cube z, cube the coefficient and multiply the exponent by 3. To find the reciprocal of w, invert the coefficient and negate the exponent of 10, then rewrite in standard form.

4

Multiply z^3 by w^{-1} and take the square root to find the numerator of the secondary intensity.

z^3 \cdot w^{-1} = 16.384 \times 10^{-3} = 1.6384 \times 10^{-2}

; therefore,

\sqrt{z^3 \cdot w^{-1}} = 1.28 \times 10^{-1} = 0.128

Multiply the coefficients and add the exponents. Take the square root of the resulting coefficient and divide the exponent of 10 by 2.

5

Divide the simplified numerator by the denominator of the secondary signal intensity.

I_2 = \frac{1.28 \times 10^{-1}}{2.0 \times 10^{-2}} = 0.64 \times 10^1 = 6.4

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

6

Determine the ratio of the primary intensity to the secondary intensity.

\frac{I_1}{I_2} = \frac{10000}{6.4} = 1562.5 = 1.5625 \times 10^3

Divide the primary signal intensity value by the secondary signal intensity value, then convert the result to standard scientific notation.

Anahtar Kavram

Simplifying algebraic operations on terms involving powers, roots, and scientific notation.

Soru 203Soru →

A baker prepares a batch of sourdough bread using flour, water, and yeast. The correct recipe requires a flour-to-water ratio of $5:3$ by weight, and a water-to-yeast ratio of $8:1$ by weight. However, when mixing the ingredients, the baker accidentally uses a flour-to-water ratio of $3:2$ by weight while keeping the correct water-to-yeast ratio of $8:1$ . If the baker used exactly $12$ ounces of yeast in this incorrect batch, how many ounces of flour must be added to the mixture to restore the correct flour-to-water ratio of $5:3$ ?

16

32

48

96

160

Cevabı ve açıklamayı göster

Cevap: $16$

Cevap

The baker must add

16

ounces of flour.

First, use the correct water-to-yeast ratio of

8:1

to find the weight of water in the batch. Since

12

ounces of yeast were used, the weight of the water is

12 \times 8 = 96

ounces. Next, determine the initial weight of flour using the incorrect flour-to-water ratio of

3:2

. The initial weight of flour is

96 \times \frac{3}{2} = 144

ounces. To restore the correct flour-to-water ratio of

5:3

, find the target weight of flour for

96

ounces of water, which is

96 \times \frac{5}{3} = 160

ounces. Finally, subtract the initial weight of flour from the target weight of flour to find the additional flour needed:

160 - 144 = 16

ounces.

Adım Adım Çözüm

1

Find the weight of water using the correct water-to-yeast ratio.

The water weight is

12 \text{ oz} \times 8 = 96 \text{ ounces}

.

The water-to-yeast ratio is

8:1

by weight, and

12

ounces of yeast were used.

2

Find the initial weight of flour in the incorrect batch.

The initial flour weight is

96 \text{ oz} \times \frac{3}{2} = 144 \text{ ounces}

.

The incorrect batch was mixed with a flour-to-water ratio of

3:2

by weight.

3

Find the target weight of flour required for the correct ratio.

The target flour weight is

96 \text{ oz} \times \frac{5}{3} = 160 \text{ ounces}

.

The correct recipe requires a flour-to-water ratio of

5:3

by weight.

4

Calculate the weight of additional flour needed.

The additional flour needed is

160 \text{ oz} - 144 \text{ oz} = 16 \text{ ounces}

.

Subtracting the initial flour weight from the target flour weight gives the amount to be added.

Anahtar Kavram

Ratios and Proportions

Soru 204Soru →

A rectangular field has a length of $1.5 \times 10^4$ meters and a width of $4.0 \times 10^3$ meters. If a tractor can mow $6.0 \times 10^5$ square meters per hour, how many hours will it take the tractor to mow the entire field?

Cevabı ve açıklamayı göster

Cevap: 100

Cevap

It will take 100 hours for the tractor to mow the entire field.

The total area of the field is the product of its length and width:

(1.5 \times 10^4) \times (4.0 \times 10^3) = 6.0 \times 10^7

square meters. Dividing this area by the tractor's mowing rate of

6.0 \times 10^5

square meters per hour gives

\frac{6.0 \times 10^7}{6.0 \times 10^5} = 1.0 \times 10^2 = 100

hours.

Adım Adım Çözüm

1

Calculate the area of the rectangular field.

Area =

6.0 \times 10^7

square meters

The area of a rectangle is found by multiplying its length by its width:

(1.5 \times 10^4 \text{ m}) \times (4.0 \times 10^3 \text{ m}) = 6.0 \times 10^7 \text{ m}^2

.

2

Calculate the time required to mow the field.

Time =

100

hours

Divide the total area by the tractor's mowing rate:

\frac{6.0 \times 10^7 \text{ m}^2}{6.0 \times 10^5 \text{ m}^2/\text{hour}} = 1.0 \times 10^2 = 100

hours.

Anahtar Kavram

Multiplication and division of numbers in scientific notation

Soru 205Soru →

A certain paint mixture requires green, yellow, and white paint in the ratio $2:3:5$ , respectively. A painter wants to create a lighter shade of this paint by doubling the amount of white paint used while keeping the amounts of green and yellow paint the same. If the painter wants to prepare a total of $30$ gallons of this new lighter paint mixture, how many gallons of white paint will be needed?

10

12

15

20

25

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20 gallons

The original ratio of green to yellow to white paint is

2:3:5

. Since the amount of white paint is doubled while green and yellow remain the same, the new ratio of green to yellow to white is

2:3:10

. The sum of the parts in the new ratio is

2 + 3 + 10 = 15

parts. The fraction of the new mixture that is white paint is

\frac{10}{15}

, which simplifies to

\frac{2}{3}

. To find the number of gallons of white paint in a

30

-gallon mixture, multiply the fraction by

30

:

\frac{2}{3} \times 30 = 20

gallons.

Adım Adım Çözüm

1

Determine the new ratio of the paint mixture.

The new ratio of green to yellow to white paint is

2:3:10

.

The original ratio is

2:3:5

, and the painter doubles the white paint component (

5 \times 2 = 10

) while green and yellow remain unchanged.

2

Calculate the total parts in the new mixture.

Total parts =

2 + 3 + 10 = 15

parts.

To find the fraction of the mixture each component represents, we must sum the ratio parts.

3

Find the fraction of the mixture that is white paint and compute the required gallons for a

30

-gallon mixture.

Gallons of white paint =

20

gallons.

White paint represents

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

of the total mixture. Multiplying this fraction by the total volume of

30

gallons gives

\frac{2}{3} \times 30 = 20

gallons.

Anahtar Kavram

Solving multi-part ratio problems where one component is scaled.

Alternatif Yöntem

Instead of using fractions of the total, you can define a constant of proportionality

x

. The new amounts of green, yellow, and white paint are

2x

,

3x

, and

10x

gallons respectively. The total volume is

2x + 3x + 10x = 30

, which simplifies to

15x = 30

, so

x = 2

. The volume of white paint needed is

10x = 10(2) = 20

gallons.

Tahmini Süre:1m 30s

Soru 206Soru →

During a basketball season, a player scored the following number of points in their first five games: $18$ , $9$ , $24$ , $14$ , and $15$ . What is the median number of points the player scored in these five games?

15

16

24

68

80

Cevabı ve açıklamayı göster

Cevap: $15$

Cevap

15

To find the median of a dataset, the values must first be arranged in numerical order. Sorting the scores

18

,

9

,

24

,

14

, and

15

from least to greatest yields:

9

,

14

,

15

,

18

, and

24

. Since there is an odd number of scores (

5

scores), the median is the middle value, which is the third number in the sorted list. The third number is

15

.

Adım Adım Çözüm

1

Arrange the given list of scores in ascending order.

The sorted list of scores is

9

,

14

,

15

,

18

, and

24

.

Before finding the median of a dataset, the values must be sorted from least to greatest.

2

Identify the middle value in the sorted list of

5

numbers.

Since there are

5

numbers, the middle value is the third number, which is

15

.

For an odd number of data points, the median is the exact middle value of the sorted list.

Anahtar Kavram

Finding the median of an odd number of data points by first sorting the dataset in ascending order.

Tahmini Süre:45s

Soru 207Soru →

A smart thermostat is programmed to reduce a home's heating energy usage. In January, the heating energy usage is reduced by $15\%$ compared to the baseline usage. In February, the usage is reduced by an additional $\frac{1}{8}$ of January's usage level. In March, the usage is reduced by $20\%$ of February's usage level. If the baseline heating energy usage was $400$ kilowatt-hours (kWh), what is the total reduction in energy usage from the baseline to the end of March, in kWh?

Cevabı ve açıklamayı göster

Cevap: 162

Cevap

The total reduction in energy usage from the baseline to the end of March is 162 kWh.

The baseline usage of

400

kWh is reduced by

15\%

in January, leaving

340

kWh. In February, a reduction of

\frac{1}{8}

of

340

kWh reduces usage by

42.5

kWh, leaving

297.5

kWh. In March, a reduction of

20\%

of

297.5

kWh reduces usage by

59.5

kWh, leaving a final usage of

238

kWh. The total reduction is the difference between the baseline and final usage levels, which is

162

kWh.

Adım Adım Çözüm

1

Calculate January's energy reduction and January's usage level.

January reduction is

60

kWh; January usage level is

340

kWh.

January's reduction is

15\%

of the

400

kWh baseline (

0.15 \times 400 = 60

kWh). Subtracting this reduction from the baseline gives January's usage level (

400 - 60 = 340

kWh).

2

Calculate February's energy reduction and February's usage level.

February reduction is

42.5

kWh; February usage level is

297.5

kWh.

February's reduction is

\frac{1}{8}

of January's usage level (

340

kWh), which is

0.125 \times 340 = 42.5

kWh. Subtracting this from January's level gives February's usage level (

340 - 42.5 = 297.5

kWh).

3

Calculate March's energy reduction and March's usage level.

March reduction is

59.5

kWh; March usage level is

238

kWh.

March's reduction is

20\%

of February's usage level (

297.5

kWh), which is

0.20 \times 297.5 = 59.5

kWh. Subtracting this from February's level gives March's usage level (

297.5 - 59.5 = 238

kWh).

4

Calculate the total energy reduction from the baseline to the end of March.

The total reduction is

162

kWh.

The total reduction can be found by adding the reductions from each of the three months (

60 + 42.5 + 59.5 = 162

kWh) or by subtracting the final usage level from the baseline (

400 - 238 = 162

kWh).

Anahtar Kavram

Applying sequential percentage and fraction reductions to changing base values.

Soru 208Soru →

A store owner marks up the wholesale price of a jacket by $\$18$ to determine its retail price. If the retail price of the jacket is $\$75$ , and $w$ represents the wholesale price in dollars, which of the following equations can be used to find $w$ ?

w - 18 = 75

18w = 75

w + 18 = 75

\frac{w}{18} = 75

w + 75 = 18

Cevabı ve açıklamayı göster

Cevap: $w + 18 = 75$

Cevap

w + 18 = 75

A markup of

\$18

means the retail price is

\$18

more than the wholesale price, which is represented by the variable

w

. Therefore, the retail price is

w + 18

. Since the retail price is given as

\$75

, the equation that represents this relationship is

w + 18 = 75

.

Adım Adım Çözüm

1

Identify the relationship between wholesale price, markup, and retail price.

The retail price is equal to the wholesale price plus the markup.

A markup increases the cost of the item from its original wholesale value.

2

Substitute the given values and variables into the relationship.

Wholesale price is

w

, markup is

18

, and retail price is

75

, giving the equation

w + 18 = 75

.

This translates the verbal description directly into an algebraic equation.

Anahtar Kavram

Translating a real-world scenario into a one-step linear equation using addition.

Tahmini Süre:45s

Soru 209Soru →

A store manager records the number of laptop computers sold each day during a 5-day work week. The daily sales, in chronological order from Monday to Friday, are 2, 10, 9, 4, and 5. What is the positive difference between the mean and the median of the number of laptops sold daily?

3

1

4

17

21

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The positive difference between the mean and the median of the daily laptop sales is 1.

To find the correct answer, we first calculate the mean of the dataset by summing the numbers (

2 + 10 + 9 + 4 + 5 = 30

) and dividing by the count (

30 / 5 = 6

). Next, we find the median by sorting the dataset (

2, 4, 5, 9, 10

) and selecting the middle value, which is

5

. The positive difference between the mean and the median is

6 - 5 = 1

.

Adım Adım Çözüm

1

Calculate the mean of the daily laptop sales.

The sum of the laptop sales is

2 + 10 + 9 + 4 + 5 = 30

. Since there are 5 days, the mean is

\frac{30}{5} = 6

.

The mean is calculated by dividing the sum of all values by the total number of values.

2

Find the median of the daily laptop sales.

Sorting the sales in ascending order gives

2, 4, 5, 9, 10

. The middle value (the 3rd number) is

5

.

The median of a set of numbers is the middle value when the numbers are arranged in order.

3

Calculate the positive difference between the mean and the median.

The positive difference is

6 - 5 = 1

.

Subtract the smaller value from the larger value to find the positive difference.

Anahtar Kavram

Calculating and comparing descriptive statistics (mean and median) of a dataset.

Soru 210Soru →

A supercomputer can perform $4.5 \times 10^{15}$ calculations per second. If a complex simulation requires a total of $1.8 \times 10^{19}$ calculations, which of the following is the number of seconds it will take the supercomputer to complete this simulation?

2.5 \times 10^{-4}

4.0 \times 10^4

4.0 \times 10^3

2.5 \times 10^4

4.0 \times 10^{33}

Cevabı ve açıklamayı göster

Cevap: $4.0 \times 10^3$

Cevap

4.0 \times 10^3

To determine the number of seconds required to complete the simulation, divide the total number of calculations needed by the speed of the supercomputer:

\frac{1.8 \times 10^{19}}{4.5 \times 10^{15}}

. Dividing the coefficients gives

\frac{1.8}{4.5} = 0.4

. Subtracting the exponents gives

10^{19 - 15} = 10^4

. This yields

0.4 \times 10^4

. To express this value in proper scientific notation, shift the decimal point one place to the right to obtain

4.0

. This shift must be balanced by subtracting 1 from the exponent, yielding

4.0 \times 10^3

.

Adım Adım Çözüm

1

Set up the division to find the total time by dividing total calculations by the rate of calculations per second.

Time =

\frac{1.8 \times 10^{19}}{4.5 \times 10^{15}}

seconds

Dividing the total work by the rate of work per unit of time gives the total time required.

2

Divide the coefficients and subtract the exponents using scientific notation division rules.

Time =

0.4 \times 10^{19 - 15} = 0.4 \times 10^4

seconds

When dividing numbers in scientific notation, divide the decimal coefficients and subtract the exponent of the divisor from the exponent of the dividend.

3

Convert the value to standard scientific notation format (

a \times 10^n

, where

1 \le |a| < 10

).

Time =

4.0 \times 10^3

seconds

To write

0.4 \times 10^4

in standard scientific notation, move the decimal point one position to the right to make the coefficient

4.0

, which requires decreasing the exponent of 10 by 1.

Anahtar Kavram

Division of numbers in scientific notation and normalization to standard form.

Soru 211Soru →

On a vertical number line representing elevation in meters, a research drone is at coordinate $d$ and a submarine is at coordinate $s$ . The coordinate of the drone is a positive integer, and the coordinate of the submarine is a negative integer. The distance between the drone and the submarine is $150$ meters. If the absolute value of the submarine's coordinate is $4$ times the coordinate of the drone, what is the coordinate of the drone?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The coordinate of the drone is 30.

The correct coordinate of 30 is found by defining the distance between the positive drone coordinate

d

and negative submarine coordinate

s

as

d - s = 150

. Since

s < 0

, its absolute value

|s|

is equal to

-s

. Substituting

-s = 4d

gives

d - (-4d) = 150

, which simplifies to

5d = 150

and yields

d = 30

.

Adım Adım Çözüm

1

Set up equations based on the problem description.

d > 0

,

s < 0

, distance

|d - s| = 150

, and

|s| = 4d

.

To translate the verbal description of elevations and distances into mathematical expressions.

2

Simplify the distance and absolute value expressions using the signs of the coordinates.

Since

d

is positive and

s

is negative,

d - s = 150

. Since

s

is negative,

|s| = -s

, so

-s = 4d

or

s = -4d

.

To eliminate absolute values based on the known signs of the variables.

3

Substitute the expression for

s

into the distance equation and solve for

d

.

d - (-4d) = 150 \Rightarrow 5d = 150 \Rightarrow d = 30

.

To solve the system of linear equations to find the drone's coordinate.

Anahtar Kavram

Using absolute value to represent distance on a number line and solving equations involving sign constraints.

Soru 212Soru →

A researcher is cataloging a collection of $T$ historic documents. She catalogs the documents at a constant rate of $r$ documents per day. After $d$ days of cataloging, the remaining documents represent a fraction, $f$ , of the entire collection. Which of the following equations correctly expresses the total number of documents in the collection, $T$ , in terms of $r$ , $d$ , and $f$ ?

T = \frac{rd}{f}

T = rd(1-f)

T = \frac{rd}{1-f}

T = \frac{r(1-f)}{d}

T = rdf

Cevabı ve açıklamayı göster

Cevap: $T = \frac{rd}{1-f}$

Cevap

The correct equation is

T = \frac{rd}{1-f}

.

The correct equation is found by identifying that the total number of documents cataloged is the rate

r

multiplied by the number of days

d

, which equals

rd

. Because the remaining fraction of the collection is

f

, the completed fraction of the collection is

1-f

. The number of completed documents,

rd

, must equal the completed fraction of the total collection,

(1-f)T

. Setting these equal gives

(1-f)T = rd

. Dividing both sides by the group

(1-f)

isolates the total collection, resulting in

T = \frac{rd}{1-f}

.

Adım Adım Çözüm

1

Calculate the total number of documents cataloged.

The number of cataloged documents is

rd

.

The total quantity completed is the constant rate

r

multiplied by the number of days

d

.

2

Determine the fraction of the collection that has been cataloged.

The cataloged fraction of the collection is

1-f

.

Since the remaining fraction is

f

, the completed fraction must be

1-f

because the sum of the completed and remaining fractions must equal 1.

3

Set up an equation relating the cataloged documents to the total collection.

(1-f)T = rd

The fraction of the total collection that is cataloged,

(1-f)T

, must equal the actual number of cataloged documents,

rd

.

4

Solve for the total number of documents,

T

.

T = \frac{rd}{1-f}

Dividing both sides of the equation by the coefficient

(1-f)

isolates

T

.

Anahtar Kavram

Translating a real-world scenario with rates and fractional parts into a solvable one-step equation.

Soru 213Soru →

In a school's robotics club, the ratio of experienced members to novice members was $3:5$ . After $4$ experienced members graduated and $12$ new novice members joined the club, the ratio of experienced members to novice members became $1:3$ . What was the total number of members in the robotics club before the graduations and new members joined?

18

30

32

48

56

Cevabı ve açıklamayı göster

Cevap: 48

Cevap

48

The correct answer is

48

. Let the initial number of experienced members be

3x

and novice members be

5x

. After

4

experienced members graduate and

12

novices join, the counts become

3x - 4

and

5x + 12

respectively. Setting up the ratio

\frac{3x-4}{5x+12} = \frac{1}{3}

and cross-multiplying yields

9x - 12 = 5x + 12

. Solving this equation gives

4x = 24

, which means

x = 6

. The initial total number of members was

3x + 5x = 8x = 8(6) = 48

.

Adım Adım Çözüm

1

Define variables for the initial number of experienced and novice members using the given ratio.

Let the number of experienced members be

3x

and the number of novice members be

5x

, where

x

is a constant. The initial total number of members is

3x + 5x = 8x

.

Using a multiplier

x

allows us to represent the parts of the ratio as actual quantities.

2

Set up an equation representing the ratio after the changes.

The new number of experienced members is

3x - 4

, and the new number of novice members is

5x + 12

. The new ratio is

\frac{3x - 4}{5x + 12} = \frac{1}{3}

.

Graduating members decreases the count by

4

, and new members joining increases the count by

12

.

3

Solve the equation for

x

.

Cross-multiplying gives

3(3x - 4) = 1(5x + 12)

, which simplifies to

9x - 12 = 5x + 12

. Subtracting

5x

and adding

12

to both sides gives

4x = 24

, so

x = 6

.

Solving for the multiplier

x

allows us to find the actual counts of members.

4

Calculate the initial total number of members.

The initial total number of members is

8x = 8(6) = 48

.

The question asks for the total number of members before the changes occurred.

Anahtar Kavram

Setting up and solving equations involving ratios and proportions from word problems.

Tahmini Süre:2m 0s

Soru 214Soru →

A jeweler creates a custom alloy by melting gold, silver, and copper in a ratio of $9:4:2$ by weight, respectively. If the jeweler uses $12$ ounces of silver to make the alloy, what is the total weight, in ounces, of the alloy produced?

20

27

33

45

90

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The correct answer is 45 ounces.

The correct answer is 45. In the gold, silver, and copper ratio of

9:4:2

, silver represents

4

parts. Since the jeweler uses

12

ounces of silver, each part is equal to

12 \div 4 = 3

ounces. The total weight consists of

9 + 4 + 2 = 15

parts. Multiplying the total parts by the weight of each part gives

15 \times 3 = 45

ounces.

Adım Adım Çözüm

1

Determine the value of one part in the ratio based on the given weight of silver.

Since silver corresponds to 4 parts in the

9:4:2

ratio, we set up

4x = 12

, which gives

x = 3

ounces per part.

Finding the unit rate per part allows us to scale up to the total parts.

2

Find the total number of parts in the alloy.

Total parts =

9 + 4 + 2 = 15

parts.

The total weight of the alloy is represented by the sum of the ratio parts of all components.

3

Calculate the total weight by multiplying the total parts by the weight per part.

Total weight =

15 \times 3 = 45

ounces.

This yields the complete weight of the gold, silver, and copper combined.

Anahtar Kavram

Solving multi-part ratio word problems by finding the unit value of a single part and scaling it to find the total amount.

Tahmini Süre:1m 0s

Soru 215Soru →

If $x = \frac{\sqrt{8^{12} \times 4^6}}{\sqrt[3]{16^9}}$ , what is the value of $\sqrt{x}$ ?

16

64

256

4,096

65,536

Cevabı ve açıklamayı göster

Cevap: 64

Cevap

64

First, convert all bases in the expression to base 2:

8^{12} = (2^3)^{12} = 2^{36}

,

4^6 = (2^2)^6 = 2^{12}

, and

16^9 = (2^4)^9 = 2^{36}

. The numerator becomes

\sqrt{2^{36} \times 2^{12}} = \sqrt{2^{48}} = 2^{24}

. The denominator becomes

\sqrt[3]{2^{36}} = 2^{12}

. Dividing the numerator by the denominator gives

x = \frac{2^{24}}{2^{12}} = 2^{12}

. Finally, taking the square root of

x

yields

\sqrt{x} = \sqrt{2^{12}} = 2^6 = 64

.

Adım Adım Çözüm

1

Express each base in the expression (

8

,

4

, and

16

) as a power of

2

.

The bases are rewritten as

8 = 2^3

,

4 = 2^2

, and

16 = 2^4

.

Converting all terms to a common base allows us to apply standard laws of exponents to simplify the expression.

2

Simplify the numerator

\sqrt{8^{12} \times 4^6}

using exponent rules.

\sqrt{(2^3)^{12} \times (2^2)^6} = \sqrt{2^{36} \times 2^{12}} = \sqrt{2^{48}} = 2^{24}

.

We multiply the exponents for the power of a power rule, add the exponents when multiplying powers with the same base, and then divide the exponent by 2 to take the square root.

3

Simplify the denominator

\sqrt[3]{16^9}

using exponent rules.

\sqrt[3]{(2^4)^9} = \sqrt[3]{2^{36}} = 2^{12}

.

We multiply the exponents for the power of a power rule and then divide the exponent by 3 to take the cube root.

4

Evaluate

x

by dividing the simplified numerator by the simplified denominator.

x = \frac{2^{24}}{2^{12}} = 2^{24 - 12} = 2^{12}

.

We subtract the exponent of the denominator from the exponent of the numerator when dividing powers with the same base.

5

Calculate the value of

\sqrt{x}

as requested in the question.

\sqrt{x} = \sqrt{2^{12}} = 2^{12/2} = 2^6 = 64

.

Taking the square root of a base to a power is equivalent to dividing the exponent by 2.

Anahtar Kavram

Applying properties of exponents and radical operations to simplify rational expressions.

Soru 216Soru →

An employee at a retail store recorded the number of hours they worked each day for one week in the table below.

Day	Hours Worked
Monday	8
Tuesday	2
Wednesday	10
Thursday	5
Friday	3
Saturday	7
Sunday	7

What is the median number of hours worked per day for this week?

5

6

7

8

36

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The median number of hours worked per day for this week is 7.

The median of a set of data is the middle value when the numbers are arranged in order. Sorting the given hours worked from least to greatest gives: 2, 3, 5, 7, 7, 8, 10. Since there are 7 data points, the middle value is the 4th value, which is 7.

Adım Adım Çözüm

1

Arrange the daily hours worked in ascending order.

The sorted list of hours is: 2, 3, 5, 7, 7, 8, 10.

Finding the median of a data set requires organizing the values from least to greatest.

2

Identify the middle value in the ordered list of 7 values.

The 4th value in the sorted list is 7.

Since the data set has an odd number of values (7), the median is the single middle value, located at the 4th position.

Anahtar Kavram

The median of a data set is the middle value when the data are arranged in numerical order.

Soru 217Soru →

A chef is preparing a salad dressing. The mixture consists of $\frac{1}{4}$ cup of olive oil, $\frac{1}{5}$ cup of balsamic vinegar, and a certain amount of lemon juice. If the lemon juice accounts for exactly $40\%$ of the total volume of the dressing, what is the total volume, in cups, of the prepared dressing?

\frac{10}{27}

\frac{5}{9}

\frac{3}{4}

\frac{17}{20}

\frac{9}{8}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{4}$

Cevap

\frac{3}{4}

To find the total volume, first calculate the sum of the volumes of olive oil and balsamic vinegar:

\frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}

cup. Because the lemon juice accounts for

40\%

of the total volume of the dressing, the remaining ingredients (olive oil and balsamic vinegar) must account for the other

60\%

(or

\frac{3}{5}

) of the total volume. Letting

T

represent the total volume, we set up the equation

\frac{3}{5} T = \frac{9}{20}

. Solving for

T

yields

T = \frac{9}{20} \times \frac{5}{3} = \frac{3}{4}

cups. Therefore, the option representing three-fourths of a cup is the correct answer.

Adım Adım Çözüm

1

Add the volumes of the two known ingredients, olive oil and balsamic vinegar.

Combined volume is

\frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}

cup (or

0.45

cup).

To find what fraction of the total volume the known ingredients represent, we first need their combined total.

2

Determine the percentage of the total volume that the olive oil and balsamic vinegar represent.

These two ingredients represent

100\% - 40\% = 60\%

(or

\frac{3}{5}

) of the total volume.

Since lemon juice accounts for

40\%

of the total volume, the rest of the ingredients must account for the remaining percentage.

3

Set up a linear equation to solve for the total volume,

T

.

\frac{3}{5} T = \frac{9}{20} \Rightarrow T = \frac{9}{20} \times \frac{5}{3} = \frac{3}{4}

cups (or

0.75

cups).

Solving the equation gives the total volume of the entire dressing.

Anahtar Kavram

Solving multi-step word problems involving fractions, decimals, and percentages by relating parts to the whole.

Alternatif Yöntem

Convert the fractions to decimals first:

\frac{1}{4} = 0.25

cups of olive oil and

\frac{1}{5} = 0.20

cups of balsamic vinegar. Their combined volume is

0.25 + 0.20 = 0.45

cups. Since lemon juice is

40\%

of the total volume, the other ingredients make up

100\% - 40\% = 60\%

of the total volume. Let the total volume be

T

. We set up the equation

0.60 T = 0.45

, which simplifies to

T = \frac{0.45}{0.60} = 0.75

cups, or

\frac{3}{4}

cups.

Tahmini Süre:1m 30s

Soru 218Soru →

Arrange the following numbers in order from least to greatest: $\left(\frac{1}{4}\right)^3$ , $\sqrt{0.0004}$ , $2.4 \times 10^{-2}$ , and $5^{-2}$ .

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order of the values from least to greatest is

\left(\frac{1}{4}\right)^3

, followed by

\sqrt{0.0004}

, then

2.4 \times 10^{-2}

, and finally

5^{-2}

.

Converting all expressions to decimals allows direct comparison:

\left(\frac{1}{4}\right)^3 = 0.015625

,

\sqrt{0.0004} = 0.02

,

2.4 \times 10^{-2} = 0.024

, and

5^{-2} = 0.04

. Arranging these decimals from smallest to largest results in the sequence

0.015625 < 0.02 < 0.024 < 0.04

.

Adım Adım Çözüm

1

Convert each of the exponential, root, and scientific notation expressions into decimal values to make comparison straightforward.

The expressions convert as follows:
-

\left(\frac{1}{4}\right)^3 = \frac{1}{64} = 0.015625

-

\sqrt{0.0004} = \sqrt{4 \times 10^{-4}} = 2 \times 10^{-2} = 0.02

-

2.4 \times 10^{-2} = 0.024

-

5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04

Expressing all values in standard decimal notation eliminates different bases and representations, allowing direct comparison.

2

Compare and arrange the resulting decimals from the smallest value to the largest value.

0.015625 < 0.02 < 0.024 < 0.04

By aligning the place values, we can see that

0.015625

(one hundredth plus fractional parts) is less than

0.02

(two hundredths), which is less than

0.024

(twenty-four thousandths), which is less than

0.04

(four hundredths).

Anahtar Kavram

Comparing numerical expressions involving negative exponents, fractional powers, roots, and scientific notation by converting them to a common decimal representation.

Soru 219Soru →

A local shipping warehouse records the number of packages shipped during each hour of a 6-hour morning shift in the table below:

Hour	Packages Shipped
Hour 1	$15$
Hour 2	$27$
Hour 3	$18$
Hour 4	$32$
Hour 5	$12$
Hour 6	$22$

What is the median number of packages shipped per hour during this shift?

20

21

22

25

29

Cevabı ve açıklamayı göster

Cevap: $20$

Cevap

The median number of packages shipped per hour is

20

.

The correct answer is

20

. To find the median, we first arrange the numbers in ascending order:

12, 15, 18, 22, 27, 32

. Since there is an even number of values (

6

), the median is the average of the two middle values,

18

and

22

. The average is

\frac{18 + 22}{2} = 20

.

Adım Adım Çözüm

1

Sort the package counts in ascending order.

The sorted dataset is

12, 15, 18, 22, 27, 32

.

Finding the median of a dataset requires the data to be ordered from least to greatest.

2

Identify the two middle numbers in the ordered dataset.

Since there is an even number of data points (

6

values), the two middle numbers are the 3rd and 4th values:

18

and

22

.

For an even number of data points, the median is the average of the two central values.

3

Calculate the average of the two middle numbers.

\frac{18 + 22}{2} = \frac{40}{2} = 20

.

The average of

18

and

22

yields the median value of

20

.

Anahtar Kavram

Calculating the median of a dataset with an even number of values.

Soru 220Soru →

At a certain high school, the ratio of the number of students in the band to the number of students in the orchestra is $3:2$ . The ratio of the number of students in the orchestra to the number of students in the choir is $4:5$ . If there are $120$ students in the band, how many students are in the choir?

64

80

100

200

225

Cevabı ve açıklamayı göster

Cevap: $100$

Cevap

100 students are in the choir

To find the number of students in the choir, we first use the ratio of band students to orchestra students (

3:2

) to determine the number of orchestra students. Since there are

120

band students, we set up the proportion

\frac{120}{\text{orchestra}} = \frac{3}{2}

, which yields

80

orchestra students. Next, we use the ratio of orchestra students to choir students (

4:5

) to find the number of choir students. Setting up the proportion

\frac{80}{\text{choir}} = \frac{4}{5}

gives

\text{choir} = 80 \times \frac{5}{4} = 100

students.

Adım Adım Çözüm

1

Use the ratio of band students to orchestra students to find the number of orchestra students.

The number of orchestra students is

80

.

Since the ratio of band to orchestra is

3:2

, and there are

120

band students, we solve the proportion

\frac{120}{\text{Orchestra}} = \frac{3}{2}

, giving

\text{Orchestra} = 120 \times \frac{2}{3} = 80

.

2

Use the ratio of orchestra students to choir students to find the number of choir students.

The number of choir students is

100

.

Since the ratio of orchestra to choir is

4:5

, and there are

80

orchestra students, we solve the proportion

\frac{80}{\text{Choir}} = \frac{4}{5}

, giving

\text{Choir} = 80 \times \frac{5}{4} = 100

.

Anahtar Kavram

Solving multi-step ratio problems by finding a common linking quantity.

Pre-Algebra

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

Alternatif Yöntem

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Alternatif Yöntem

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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