Pre-Algebra Practice Questions — ACT — Page 10

Question 181Question →

A video streaming service has three subscription tiers: Basic, Standard, and Premium. Originally, $\frac{2}{5}$ of the subscribers are on the Basic tier and $45\%$ are on the Standard tier. During a promotional campaign, $\frac{1}{4}$ of the Basic subscribers upgrade to the Standard tier, and $20\%$ of the original Standard subscribers upgrade to the Premium tier. If no other subscription changes occur, what percent of the total subscribers are now on the Premium tier?

24\%

26\%

35\%

65\%

9\%

Show answer & explanation

Answer: $24\%$

Answer

The correct answer is

24\%

of the total subscribers.

To find the new Premium subscriber percentage, we first convert the basic tier fraction to a percent:

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

. Since the Standard tier is

45\%

, the original Premium tier must contain the remaining subscribers, which is

100\% - 40\% - 45\% = 15\%

of the total. The upgrade represents

20\%

of the original

45\%

Standard subscribers, which is

0.20 \times 45\% = 9\%

of the total subscribers. Adding this

9\%

upgrade to the original

15\%

Premium percentage results in a final percentage of

24\%

.

Step-by-Step Solution

1

Determine the original subscriber percentages for all three tiers.

Basic:

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

. Standard:

45\%

. Premium:

100\% - (40\% + 45\%) = 15\%

.

Establishing the starting percentages allows us to track changes relative to the total subscriber population.

2

Calculate the percentage of total subscribers upgrading from Standard to Premium.

20\%

of the original

45\%

Standard subscribers =

0.20 \times 45\% = 9\%

of the total subscribers.

Since the upgrade rate is relative to the Standard tier, we multiply the rate by the Standard tier's share of the total population.

3

Calculate the final Premium percentage by adding the upgrade percentage to the original Premium percentage.

15\% + 9\% = 24\%

.

The final Premium percentage is the sum of the original Premium percentage and the percentage of subscribers who upgraded from Standard.

Key Concept

Calculating percentages of subpopulations and translating them to percentages of the entire population.

Question 182Question →

An astronomical unit (AU) is a unit of length equal to the average distance from the Earth to the Sun, approximately $1.5 \times 10^8$ kilometers. A scientific probe travels through space at a constant speed of $2.5 \times 10^4$ meters per second. How many hours would it take the probe to travel a distance of $3.6 \times 10^2$ AU?

6.0 \times 10^2

6.0 \times 10^5

6.0 \times 10^{11}

3.6 \times 10^7

3.75 \times 10^{14}

Show answer & explanation

Answer: $6.0 \times 10^5$

Answer

6.0 \times 10^5

hours

To find the travel time, we first determine the total distance in meters. Since

1\text{ AU} = 1.5 \times 10^8\text{ km}

and

1\text{ km} = 10^3\text{ m}

,

1\text{ AU} = 1.5 \times 10^{11}\text{ m}

. Thus, a distance of

3.6 \times 10^2\text{ AU}

equals

(3.6 \times 10^2) \times (1.5 \times 10^{11}) = 5.4 \times 10^{13}\text{ m}

. Dividing this distance by the speed of

2.5 \times 10^4\text{ m/s}

yields the travel time in seconds:

\frac{5.4 \times 10^{13}}{2.5 \times 10^4} = 2.16 \times 10^9\text{ s}

. Finally, converting seconds to hours by dividing by

3.6 \times 10^3\text{ s/h}

gives

\frac{2.16 \times 10^9}{3.6 \times 10^3} = 0.6 \times 10^6 = 6.0 \times 10^5\text{ hours}

.

Step-by-Step Solution

1

Convert the total distance from astronomical units (AU) to kilometers, and then to meters.

Distance =

3.6 \times 10^2 \text{ AU} \times 1.5 \times 10^8 \text{ km/AU} = 5.4 \times 10^{10} \text{ km} = 5.4 \times 10^{13} \text{ m}

.

Since the speed is given in meters per second, the total distance must be in meters to keep units consistent.

2

Calculate the travel time in seconds by dividing the distance in meters by the speed in meters per second.

Time in seconds =

\frac{5.4 \times 10^{13} \text{ m}}{2.5 \times 10^4 \text{ m/s}} = 2.16 \times 10^9 \text{ s}

.

Time is equal to distance divided by speed (

t = \frac{d}{s}

).

3

Convert the travel time from seconds to hours by dividing by

3.6 \times 10^3

(the number of seconds in one hour).

Time in hours =

\frac{2.16 \times 10^9}{3.6 \times 10^3} = 6.0 \times 10^5 \text{ hours}

.

Dividing the total seconds by

3,600

yields the time in hours.

Key Concept

Operations with scientific notation and dimensional analysis involving distance, rate, and time.

Alternative Method

Convert the speed to astronomical units per hour first, then divide the distance in astronomical units by this speed.

Estimated Time:2m 0s

Question 183Question →

Arrange the following four mathematical expressions in order from least to greatest value:

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is

S

,

P

,

R

, and

Q

, which correspond to the decimal values

0.16 < 0.25 < 0.28 < 0.3

.

Evaluating each term yields decimal values:

S = 0.16

,

P = 0.25

,

R = 0.28

, and

Q = 0.3

. Ordering these from least to greatest gives

0.16 < 0.25 < 0.28 < 0.3

, which corresponds to the sequence

S, P, R, Q

.

Step-by-Step Solution

1

Convert the fractional squared expression

S

to a decimal.

S = \left(\frac{2}{5}\right)^2 = \frac{4}{25} = 0.16

Squaring a fraction involves squaring both the numerator and denominator, then converting the resulting fraction to decimal form.

2

Convert the negative exponent expression

P

to a decimal.

P = 4^{-1} = \frac{1}{4} = 0.25

A negative exponent represents the reciprocal of the base raised to the positive power.

3

Convert the scientific notation expression

R

to a decimal.

R = 2.8 \times 10^{-1} = 0.28

Multiplying by

10^{-1}

shifts the decimal point one position to the left.

4

Convert the square root expression

Q

to a decimal.

Q = \sqrt{0.09} = 0.3

Finding the square root of a decimal less than

1

results in a value larger than the original decimal, since

0.3 \times 0.3 = 0.09

.

5

Compare the resulting decimal values to order them from least to greatest.

0.16 < 0.25 < 0.28 < 0.3

, so the order is

S < P < R < Q

.

Ordering the converted decimal values yields the correct final arrangement.

Key Concept

Converting exponential, radical, and scientific notation expressions to decimals to compare and order their values.

Question 184Question →

A bookstore has a sale where the price of any paperback book is reduced by $4$ dollars. If the sale price of a paperback book is $12$ dollars, and its original price is $p$ dollars, which of the following equations can be used to find $p$ ?

p - 4 = 12

p + 4 = 12

4 - p = 12

4p = 12

\frac{p}{4} = 12

Show answer & explanation

Answer: $p - 4 = 12$

Answer

The equation

p - 4 = 12

The original price of the book is

p

dollars. Since the price is reduced by

4

dollars, we subtract

4

from

p

, which is written as

p - 4

. Since this resulting sale price is equal to

12

dollars, the correct equation representing the relationship is

p - 4 = 12

.

Step-by-Step Solution

1

Identify the variable and the mathematical values given in the scenario.

The original price is represented by the variable

p

. The price reduction is

4

dollars, and the final sale price is

12

dollars.

This sets up the translation of the word problem into algebraic terms.

2

Translate the phrase 'reduced by

4

dollars' into an algebraic expression.

A reduction of

4

dollars from the original price

p

is written as

p - 4

.

A reduction or discount represents subtraction from the starting amount.

3

Equate the expression representing the sale price to the given sale price value.

The equation is

p - 4 = 12

.

Since the sale price is given as

12

dollars, the expression for the reduced price must equal

12

.

Key Concept

Translating real-world discount scenarios into one-step linear equations.

Estimated Time:45s

Question 185Question →

A manufacturing plant operates two types of machines, Type A and Type B. The ratio of the number of Type A machines to Type B machines is $3:5$ . Each Type A machine produces widgets at a constant rate that is $60\%$ faster than the constant rate of a Type B machine. When all machines of both types are operating simultaneously, they produce a total of $98$ widgets per hour. If the plant increases the number of Type A machines by $50\%$ and decreases the number of Type B machines by $20\%$ , what is the total number of widgets the new setup will produce in $3$ hours?

112

246

312

336

382

Show answer & explanation

Answer: 336

Answer

336

The correct answer is 336 because setting up the initial production equation based on the ratios gives a compound constant of

xr = 10

. Applying the percentage adjustments yields a new hourly production rate of 112 widgets. Multiplying this hourly rate by the specified 3 hours gives a total of 336 widgets.

Step-by-Step Solution

1

Represent the machine counts and rates using variables based on the given ratios.

Let the number of Type A machines be

3x

and the number of Type B machines be

5x

. Let the rate of a Type B machine be

r

widgets/hour. The rate of a Type A machine is

1.6r

widgets/hour.

This establishes algebraic expressions for the quantities in terms of common multipliers

x

and

r

.

2

Set up an equation for the initial total hourly production and solve for

xr

.

(3x)(1.6r) + (5x)(r) = 98 \implies 4.8xr + 5xr = 98 \implies 9.8xr = 98 \implies xr = 10

.

This allows us to find the value of the joint constant factor

xr

needed for subsequent calculations.

3

Determine the new machine counts after the percentage changes.

New Type A count:

3x \times 1.50 = 4.5x

. New Type B count:

5x \times 0.80 = 4x

.

This applies the 50% increase to Type A and the 20% decrease to Type B machine counts.

4

Calculate the new total hourly production rate using the value of

xr

.

New hourly rate:

(4.5x)(1.6r) + (4x)(r) = 7.2xr + 4xr = 11.2xr

. Substituting

xr = 10

yields

11.2 \times 10 = 112

widgets/hour.

This finds the rate at which widgets are produced in the new configuration.

5

Multiply the new hourly rate by the specified time of 3 hours.

Total production:

112 \text{ widgets/hour} \times 3 \text{ hours} = 336

widgets.

This yields the final total quantity of widgets produced over the 3-hour duration.

Key Concept

Solving compound ratio and rate problems with percentage changes

Alternative Method

Instead of variables, you can plug in convenient numbers that satisfy the ratios. Suppose there are initially

30

machines of Type A and

50

machines of Type B. Let a Type B machine produce at a rate of

1

widget/hour, so a Type A machine produces

1.6

widgets/hour. The total initial hourly rate is

30(1.6) + 50(1) = 48 + 50 = 98

widgets/hour, which matches the prompt. Increasing Type A by 50% gives

45

machines, and decreasing Type B by 20% gives

40

machines. The new hourly rate is

45(1.6) + 40(1) = 72 + 40 = 112

widgets/hour. For 3 hours, this gives

112 \times 3 = 336

widgets.

Estimated Time:3m 0s

Question 186Question →

Arrange the following mathematical expressions in order from least to greatest value:

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest value is

(2.5 \times 10^1)^2

, followed by

\sqrt[3]{3.43 \times 10^8}

, then

\sqrt{6.4 \times 10^5}

, and finally

\frac{5.4 \times 10^6}{6 \times 10^3}

.

Each expression is simplified to a standard integer representation:

(2.5 \times 10^1)^2 = 625

,

\sqrt[3]{3.43 \times 10^8} = 700

,

\sqrt{6.4 \times 10^5} = 800

, and

\frac{5.4 \times 10^6}{6 \times 10^3} = 900

. Comparing these integers from smallest to largest yields the correct sequence.

Step-by-Step Solution

1

Evaluate the expression

\sqrt{6.4 \times 10^5}

by converting the radicand to a standard square value.

\sqrt{64 \times 10^4} = 8 \times 10^2 = 800

Converting the base to 64 and reducing the exponent of 10 to an even number allows easy extraction of the square root.

2

Evaluate the expression

(2.5 \times 10^1)^2

by simplifying the terms inside the parentheses first.

(25)^2 = 625

Multiplying 2.5 by 10 simplifies the base to 25 before applying the exponent.

3

Evaluate the expression

\frac{5.4 \times 10^6}{6 \times 10^3}

using rules of scientific division.

0.9 \times 10^3 = 900

Dividing the coefficients (

5.4 \div 6 = 0.9

) and subtracting the exponent in the denominator from the numerator (

10^{6-3} = 10^3

) yields the result.

4

Evaluate the expression

\sqrt[3]{3.43 \times 10^8}

by converting the radicand to a standard cube value.

\sqrt[3]{343 \times 10^6} = 7 \times 10^2 = 700

Converting the base to 343 and adjusting the exponent to 6 (a multiple of 3) allows easy extraction of the cube root.

5

Compare the evaluated integers to determine the correct order from least to greatest.

625 < 700 < 800 < 900

Comparing the simplified values reveals that the correct sequence is

(2.5 \times 10^1)^2

,

\sqrt[3]{3.43 \times 10^8}

,

\sqrt{6.4 \times 10^5}

, and

\frac{5.4 \times 10^6}{6 \times 10^3}

.

Key Concept

Simplification of exponential, radical, and scientific notation expressions to comparable decimal values.

Alternative Method

Instead of converting all expressions to standard numbers, convert them all to scientific notation with a base power of

10^2

(i.e.,

6.25 \times 10^2

,

7.0 \times 10^2

,

8.0 \times 10^2

, and

9.0 \times 10^2

) to compare their coefficients directly.

Estimated Time:2m 0s

Question 187Question →

Four different brands of a sports drink contain different proportions of electrolytes by weight. The proportions for each brand are listed below:

* Brand P: $\frac{5}{12}$ of its weight is electrolytes
* Brand Q: $41.8\%$ of its weight is electrolytes
* Brand R: $0.416$ of its weight is electrolytes
* Brand S: $\frac{13}{30}$ of its weight is electrolytes

Arrange the brands in order from the least proportion of electrolytes to the greatest proportion of electrolytes.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest proportion of electrolytes is Brand R, Brand P, Brand Q, and Brand S.

To compare the values, converting each to a decimal is the most reliable strategy. Brand R is already a decimal (

0.416

). Converting Brand P yields

\frac{5}{12} \approx 0.4167

. Converting Brand Q yields

41.8\% = 0.418

. Converting Brand S yields

\frac{13}{30} \approx 0.4333

. Comparing these decimals reveals that

0.416 < 0.4167 < 0.418 < 0.4333

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

Step-by-Step Solution

1

Identify and list all values given for each brand.

Brand P:

\frac{5}{12}

, Brand Q:

41.8\%

, Brand R:

0.416

, Brand S:

\frac{13}{30}

.

This establishes the raw numbers that need comparison.

2

Convert each value into a decimal format to allow direct comparison, rounding to four decimal places where necessary.

Brand R:

0.4160

; Brand P:

5 \div 12 \approx 0.4167

; Brand Q:

41.8\% = 0.4180

; Brand S:

13 \div 30 \approx 0.4333

.

Converting to decimals standardizes the formats and makes close comparisons straightforward.

3

Compare the standardized decimal values from smallest to largest.

0.4160 < 0.4167 < 0.4180 < 0.4333

, which corresponds to the order: Brand R, Brand P, Brand Q, Brand S.

Ordering the values correctly identifies the brands from the least proportion of electrolytes to the greatest.

Key Concept

Comparing and ordering rational numbers by converting fractions, decimals, and percents into a single consistent format.

Alternative Method

Instead of converting to decimals, all values can be converted to percentages: Brand R is

41.6\%

, Brand P is

\frac{5}{12} \times 100\% \approx 41.67\%

, Brand Q is

41.8\%

, and Brand S is

\frac{13}{30} \times 100\% \approx 43.33\%

. Comparing these values gives

41.6\% < 41.67\% < 41.8\% < 43.33\%

.

Estimated Time:1m 30s

Question 188Question →

A set of cards is numbered consecutively from $1$ to $N$ . If exactly $12$ of these cards have a number that is a multiple of both $6$ and $8$ , what is the greatest possible value of $N$ ?

Show answer & explanation

Answer: 311

Answer

The greatest possible value of

N

is

311

.

The correct value is 311. Since the numbers must be multiples of both 6 and 8, they must be multiples of their least common multiple, which is 24. The twelfth multiple of 24 is 288, and the thirteenth multiple is 312. To have exactly 12 such multiples, the maximum number N must be at least 288 but strictly less than 312, meaning the largest integer value is 311.

Step-by-Step Solution

1

Find the least common multiple (LCM) of 6 and 8.

The LCM of 6 and 8 is 24.

Any number that is a multiple of both 6 and 8 must be a multiple of their least common multiple.

2

Calculate the 12th and 13th multiples of 24.

The 12th multiple is

12 \times 24 = 288

and the 13th multiple is

13 \times 24 = 312

.

To have exactly 12 multiples in the set, the set must include the 12th multiple but exclude the 13th multiple.

3

Determine the maximum value of

N

such that 312 is not included.

N = 311

.

The largest integer less than 312 is 311. If

N

were 312 or greater, the set would contain 13 or more multiples.

Key Concept

Least Common Multiple and Divisibility Properties

Question 189Question →

A runner completed a race in $45$ minutes. If this time is $12$ minutes faster than their previous race time, $t$ , which of the following equations can be used to find the previous race time?

t - 12 = 45

t + 12 = 45

12 - t = 45

12t = 45

\frac{t}{12} = 45

Show answer & explanation

Answer: $t - 12 = 45$

Answer

The equation

t - 12 = 45

correctly models the relationship between the previous and current race times.

The equation

t - 12 = 45

is correct because 'faster than' in running means a shorter duration of time. Thus, the previous time

t

must be

12

minutes greater than

45

minutes, which is algebraically expressed as

t - 12 = 45

(or

t = 45 + 12

).

Step-by-Step Solution

1

Identify the relationship between the current time and the previous time

t

.

The current time is

45

minutes, which is

12

minutes faster than

t

.

This establishes that the previous time

t

is larger than the current time of

45

minutes.

2

Set up the algebraic equation based on the time difference.

t - 12 = 45

Subtracting the difference of

12

minutes from the slower previous time

t

yields the faster current time of

45

minutes.

Key Concept

Translating verbal descriptions of differences into one-step subtraction equations

Estimated Time:45s

Question 190Question →

Simplify the four mathematical expressions below. How should these expressions be ordered from least to greatest based on their simplified values?

* Expression A: $\sqrt{2.25 \times 10^4} \times \left(2.0 \times 10^{-1}\right)^{-2}$
* Expression B: $\frac{\left(3.0 \times 10^2\right)^3}{6.0 \times 10^3}$
* Expression C: $\sqrt[3]{2.7 \times 10^7} \times \left(4.0 \times 10^{-2}\right)^{-1/2}$
* Expression D: $\left(6.4 \times 10^5\right)^{1/2} \times 5.0 \times 10^0$

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Expression C, Expression A, Expression D, Expression B

The correct order is determined by fully simplifying each expression: Expression C simplifies to

1,500

, Expression A simplifies to

3,750

, Expression D simplifies to

4,000

, and Expression B simplifies to

4,500

. Arranging these values in ascending order gives the correct sequence starting with Expression C, followed by Expression A, then Expression D, and ending with Expression B.

Step-by-Step Solution

1

Simplify Expression C:

\sqrt[3]{2.7 \times 10^7} \times \left(4.0 \times 10^{-2}\right)^{-1/2}

C = 1,500

First, rewrite

2.7 \times 10^7

as

27 \times 10^6

to compute the cube root easily:

\sqrt[3]{27 \times 10^6} = 3 \times 10^2 = 300

. Second, evaluate the term with the rational exponent:

\left(4.0 \times 10^{-2}\right)^{-1/2} = \frac{1}{\sqrt{4.0 \times 10^{-2}}} = \frac{1}{2.0 \times 10^{-1}} = \frac{1}{0.2} = 5

. Finally, multiply the two parts:

300 \times 5 = 1,500

.

2

Simplify Expression A:

\sqrt{2.25 \times 10^4} \times \left(2.0 \times 10^{-1}\right)^{-2}

A = 3,750

First, calculate the square root:

\sqrt{2.25 \times 10^4} = \sqrt{2.25} \times \sqrt{10^4} = 1.5 \times 10^2 = 150

. Second, evaluate the term with the negative exponent:

\left(2.0 \times 10^{-1}\right)^{-2} = (0.2)^{-2} = \frac{1}{0.04} = 25

. Finally, multiply the two simplified values:

150 \times 25 = 3,750

.

3

Simplify Expression D:

\left(6.4 \times 10^5\right)^{1/2} \times 5.0 \times 10^0

D = 4,000

First, simplify the square root by rewriting the radicand:

\left(6.4 \times 10^5\right)^{1/2} = \sqrt{64 \times 10^4} = 8 \times 10^2 = 800

. Second, evaluate

5.0 \times 10^0 = 5.0 \times 1 = 5

. Finally, multiply the results:

800 \times 5 = 4,000

.

4

Simplify Expression B:

\frac{\left(3.0 \times 10^2\right)^3}{6.0 \times 10^3}

B = 4,500

First, raise the numerator to the power of three:

\left(3.0 \times 10^2\right)^3 = 3^3 \times \left(10^2\right)^3 = 27 \times 10^6

. Next, divide by the denominator:

\frac{27 \times 10^6}{6.0 \times 10^3} = \frac{27}{6.0} \times 10^3 = 4.5 \times 10^3 = 4,500

.

5

Compare the simplified values of the four expressions.

1,500 < 3,750 < 4,000 < 4,500

, which corresponds to Expression C, Expression A, Expression D, then Expression B.

Sorting the values in ascending numerical order reveals the correct order from least to greatest.

Key Concept

Simplifying complex numeric expressions containing combinations of square roots, cube roots, negative exponents, fractional exponents, and scientific notation.

Estimated Time:3m 0s

Question 191Question →

A construction crew of $12$ workers can build $2$ identical storage sheds in $8$ days, working $6$ hours per day. To complete a new project of building $3$ of these same storage sheds in $6$ days, how many additional workers, working at the same rate, must be hired if they will work $4$ hours per day?

4

12

16

24

36

Show answer & explanation

Answer: 24

Answer

The correct answer is 24 additional workers.

The correct answer is 24 additional workers. First, calculate the rate of work:

12

workers working

6

hours per day for

8

days is

12 \times 8 \times 6 = 576

total worker-hours to complete

2

sheds, which equals

288

worker-hours per shed. For the new project of

3

sheds, the total worker-hours needed is

3 \times 288 = 864

hours. Under the new schedule of

6

days at

4

hours per day, each worker will work

6 \times 4 = 24

hours. Thus, the total number of workers required is

864 / 24 = 36

workers. Subtracting the original crew of

12

workers, the crew needs to hire

36 - 12 = 24

additional workers.

Step-by-Step Solution

1

Calculate the total worker-hours required to build one storage shed.

288 worker-hours per shed

A crew of

12

workers working for

8

days at

6

hours per day contributes a total of

12 \times 8 \times 6 = 576

worker-hours to build

2

sheds. Dividing

576

by

2

gives

288

worker-hours per shed.

2

Determine the total worker-hours required for the new project of building 3 sheds.

864 worker-hours

Since each shed requires

288

worker-hours, building

3

sheds will require

3 \times 288 = 864

worker-hours.

3

Calculate the number of hours each worker will contribute under the new schedule.

24 hours per worker

On the new schedule, workers work for

6

days at

4

hours per day, which equals

6 \times 4 = 24

hours per worker.

4

Calculate the total number of workers required and the number of additional workers to hire.

36 total workers, which means 24 additional workers

Dividing the total required hours (

864

) by the hours per worker (

24

) gives

36

total workers. Subtracting the initial crew of

12

workers yields

36 - 12 = 24

additional workers.

Key Concept

Compound rates and proportional relationships

Alternative Method

We can set up a compound proportion equation where

\frac{W_1 \cdot D_1 \cdot H_1}{S_1} = \frac{W_2 \cdot D_2 \cdot H_2}{S_2}

. Substituting the known values:

\frac{12 \cdot 8 \cdot 6}{2} = \frac{W_2 \cdot 6 \cdot 4}{3}

. Simplifying both sides gives

288 = 8 W_2

, which yields

W_2 = 36

. The number of additional workers is

36 - 12 = 24

.

Estimated Time:1m 30s

Question 192Question →

Four prototype solar cells are tested for their efficiency in converting solar energy into electricity. The efficiency of each solar cell is recorded as follows:

* Cell W: $\frac{4}{25}$ of the incoming energy is converted.
* Cell X: $16.5\%$ of the incoming energy is converted.
* Cell Y: $0.1605$ of the incoming energy is converted.
* Cell Z: $\frac{13}{80}$ of the incoming energy is converted.

Based on this data, how should these solar cells be ordered from least efficient to most efficient?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ordering from least efficient to most efficient is Cell W, Cell Y, Cell Z, and Cell X.

Converting all efficiency values to decimals facilitates comparison. Cell W is

\frac{4}{25} = 0.1600

. Cell Y is

0.1605

. Cell Z is

\frac{13}{80} = 0.1625

. Cell X is

16.5\% = 0.1650

. Comparing these decimals shows that

0.1600 < 0.1605 < 0.1625 < 0.1650

, which matches the order: Cell W, Cell Y, Cell Z, Cell X.

Step-by-Step Solution

1

Convert the efficiency of Cell W to a decimal.

\frac{4}{25} = 0.16 = 0.1600

To easily compare the values, convert the fraction to a decimal by dividing the numerator by the denominator.

2

Convert the efficiency of Cell X to a decimal.

16.5\% = 0.165 = 0.1650

Convert the percentage to a decimal by dividing by 100.

3

Convert the efficiency of Cell Z to a decimal.

\frac{13}{80} = 0.1625

Divide 13 by 80 to obtain its decimal representation.

4

Compare all four decimal values:

0.1600

(Cell W),

0.1650

(Cell X),

0.1605

(Cell Y), and

0.1625

(Cell Z).

0.1600 < 0.1605 < 0.1625 < 0.1650

Align the decimals by place value to determine their correct ascending order.

Key Concept

Comparing and ordering fractions, decimals, and percentages by converting them to a common format (decimals).

Estimated Time:1m 30s

Question 193Question →

On a standard number line, point $A$ has coordinate $-15$ and point $B$ has coordinate $17$ . Point $C$ is located to the right of point $B$ such that the distance between $A$ and $C$ is exactly $3$ times the distance between $B$ and $C$ . What is the coordinate of point $C$ ?

Show answer & explanation

Answer: 33

Answer

The coordinate of point

C

is 33.

The correct coordinate is found by setting up the distance equation for point

C

(with coordinate

c > 17

) relative to

A(-15)

and

B(17)

. The distance

AC

is

c - (-15) = c + 15

, and the distance

BC

is

c - 17

. Setting

c + 15 = 3(c - 17)

and solving yields

c = 33

, which is to the right of

B

.

Step-by-Step Solution

1

Define the variable for the coordinate of point C and write the expressions for distances.

Let

c

be the coordinate of point

C

. The distance between

A

and

C

is

|c - (-15)| = |c + 15|

, and the distance between

B

and

C

is

|c - 17|

. Since point

C

is to the right of point

B

(which is at

17

), we know

c > 17

, so

|c + 15| = c + 15

and

|c - 17| = c - 17

.

To set up an algebraic equation representing the physical distance relations on the number line.

2

Set up the equation using the given relationship.

The equation is

c + 15 = 3(c - 17)

.

The problem states the distance between

A

and

C

is

3

times the distance between

B

and

C

.

3

Solve the equation for

c

.

c + 15 = 3c - 51 \implies 15 + 51 = 3c - c \implies 66 = 2c \implies c = 33

.

To find the coordinate of point

C

.

Key Concept

Calculating distances between points on a number line using absolute value and solving the resulting equations.

Alternative Method

Use geometric visualization: The distance from

A(-15)

to

B(17)

is

17 - (-15) = 32

units. Since point

C

lies to the right of

B

, the distance

AC

is the sum of

AB

and

BC

. Therefore,

AC = 32 + BC

. We are given that

AC = 3 \times BC

. Substituting this gives

32 + BC = 3 \times BC \implies 2 \times BC = 32 \implies BC = 16

. Since

C

is 16 units to the right of

B(17)

, its coordinate is

17 + 16 = 33

.

Estimated Time:1m 30s

Question 194Question →

A laboratory sample contains $1.6 \times 10^{17}$ atoms of a radioactive isotope. The sample decays such that at the end of each 6-hour interval, the number of remaining atoms is equal to the square root of the number of atoms that were present at the beginning of that interval. Which of the following is the number of atoms remaining in the sample at the end of 12 hours?

4 \times 10^4

2 \times 10^{4.25}

2 \times 10^4

4 \times 10^8

4 \times 10^{16}

Show answer & explanation

Answer: $2 \times 10^4$

Answer

2 \times 10^4

The correct option is

2 \times 10^4

. To find the number of remaining atoms after 12 hours, we must apply the square root operation twice, since 12 hours consists of two 6-hour intervals. First, rewrite the initial number of atoms as

16 \times 10^{16}

to make the exponent even. After 6 hours, the number of atoms is

\sqrt{16 \times 10^{16}} = 4 \times 10^8

. After another 6 hours, we take the square root again:

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

.

Step-by-Step Solution

1

Determine the number of 6-hour intervals in a 12-hour period.

There are

12 \div 6 = 2

intervals.

Since the population is reduced to its square root every 6 hours, we must apply the square root operation twice over a 12-hour period.

2

Rewrite the initial number of atoms in a form that simplifies taking the square root.

1.6 \times 10^{17} = 16 \times 10^{16}

Converting the coefficient to a perfect square and the exponent to an even number allows us to easily compute the square root without a calculator.

3

Calculate the number of atoms remaining after the first 6-hour interval by taking the square root of the initial value.

\sqrt{16 \times 10^{16}} = \sqrt{16} \times \sqrt{10^{16}} = 4 \times 10^8

The square root of a product is the product of the square roots, and the square root of

10^{16}

is

10^{16 \cdot 0.5} = 10^8

.

4

Calculate the number of atoms remaining after the second 6-hour interval (12 hours total) by taking the square root of the value at the end of the first interval.

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

Applying the square root operation to the intermediate quantity of

4 \times 10^8

yields the final remaining atoms after the full 12 hours.

Key Concept

Applying square root operations to expressions written in scientific notation.

Alternative Method

We can compute the overall decay multiplier first. Taking the square root twice is equivalent to raising the initial quantity to the power of

\frac{1}{4}

. Thus, the final quantity is

(1.6 \times 10^{17})^{1/4} = (16 \times 10^{16})^{1/4} = 16^{1/4} \times (10^{16})^{1/4} = 2 \times 10^4

.

Estimated Time:2m 0s

Question 195Question →

A youth soccer league has a group of players. The players can be divided into equal-sized teams of either 12 or 18, with no players left over. However, if they are divided into teams of 15, there are exactly 3 players left over. What is the least possible number of players in the league?

3

36

72

108

180

Show answer & explanation

Answer: 108

Answer

The least possible number of players in the league is 108.

To satisfy the condition that players can be divided into teams of 12 or 18 with no remainders, the total number of players must be a multiple of their least common multiple (LCM). The prime factorization of 12 is

2^2 \times 3

, and for 18 it is

2 \times 3^2

, so their LCM is

2^2 \times 3^2 = 36

. Next, we evaluate the positive multiples of 36 to find the smallest one that leaves a remainder of 3 when divided by 15. Testing 36:

36 = 15(2) + 6

(remainder 6). Testing 72:

72 = 15(4) + 12

(remainder 12). Testing 108:

108 = 15(7) + 3

(remainder 3). Thus, 108 satisfies all conditions.

Step-by-Step Solution

1

Find the least common multiple (LCM) of 12 and 18.

The LCM of 12 and 18 is 36.

Since the players can be divided into teams of 12 or 18 without remainders, the total number of players must be a multiple of both 12 and 18, which means it must be a multiple of their LCM.

2

List the positive multiples of 36.

The multiples are 36, 72, 108, 144, 180, and so on.

To find the least possible number of players, we need to test the multiples of 36 in increasing order.

3

Test the multiples of 36 to find which one leaves a remainder of 3 when divided by 15.

Testing 36 gives a remainder of 6; testing 72 gives a remainder of 12; testing 108 gives a remainder of 3.

Dividing 108 by 15 yields 7 with a remainder of 3, satisfying all the conditions of the problem.

Key Concept

Using the least common multiple (LCM) to satisfy multiple divisibility and remainder constraints.

Estimated Time:1m 30s

Question 196Question →

An art teacher divides a container of clay equally among $6$ students. If each student receives $15$ ounces of clay, and $c$ represents the total number of ounces of clay in the container, which of the following equations can be used to find $c$ ?

6c = 15

c - 6 = 15

\frac{c}{6} = 15

c + 6 = 15

\frac{6}{c} = 15

Show answer & explanation

Answer: $\frac{c}{6} = 15$

Answer

\frac{c}{6} = 15

The equation representing the total amount of clay divided by the number of students equals the amount of clay per student is correct. Since the teacher divides the clay

c

equally among

6

students, the expression

\frac{c}{6}

represents the amount each student gets, which is given as

15

ounces. Therefore, the correct equation is

\frac{c}{6} = 15

.

Step-by-Step Solution

1

Identify the relationship between the total amount of clay, the number of students, and the amount per student.

The total amount of clay,

c

, divided by the number of students,

6

, must equal the amount of clay each student receives.

Dividing a total quantity equally among a set number of groups is represented by division.

2

Write the relationship as an algebraic equation.

\frac{c}{6} = 15

Substitute the given values and variables into the division relationship.

Key Concept

Translating word problems into one-step algebraic division equations

Estimated Time:45s

Question 197Question →

The volume of a spherical cell is given by the formula $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the cell. If a certain spherical cell has a radius of $r = 3.0 \times 10^{-6}$ meters, what is its volume in cubic meters, written in scientific notation?

1.2\pi \times 10^{-17}

1.2\pi \times 10^{-11}

3.6\pi \times 10^{-2}

3.6\pi \times 10^{-17}

6.4\pi \times 10^{-17}

Show answer & explanation

Answer: $3.6\pi \times 10^{-17}$

Answer

3.6\pi \times 10^{-17}

To find the volume, substitute the radius

r = 3.0 \times 10^{-6}

into the formula

V = \frac{4}{3}\pi r^3

. First, cube the radius:

(3.0 \times 10^{-6})^3 = 27 \times 10^{-18}

. Next, multiply this value by

\frac{4}{3}\pi

to get

36\pi \times 10^{-18}

. Finally, convert this value to standard scientific notation by rewriting it as

3.6\pi \times 10^{-17}

.

Step-by-Step Solution

1

Substitute the radius

r = 3.0 \times 10^{-6}

meters into the sphere volume formula

V = \frac{4}{3}\pi r^3

.

V = \frac{4}{3}\pi (3.0 \times 10^{-6})^3

This sets up the expression for calculating the volume.

2

Cube the term inside the parentheses using the power of a product and power of a power rules.

(3.0 \times 10^{-6})^3 = 3.0^3 \times (10^{-6})^3 = 27 \times 10^{-18}

This simplifies the radius term before performing the remaining operations.

3

Multiply the simplified term by

\frac{4}{3}\pi

.

V = \frac{4}{3}\pi \times 27 \times 10^{-18} = 36\pi \times 10^{-18}

This performs the fractional multiplication to find the volume.

4

Convert the volume into standard scientific notation by rewriting

36

as

3.6 \times 10^1

and combining the exponents.

36\pi \times 10^{-18} = 3.6\pi \times 10^{-17}

Standard scientific notation requires the coefficient to be between 1 and 10.

Key Concept

Applying laws of exponents and scientific notation rules to evaluate formulas.

Question 198Question →

A car travels at a constant speed of $r$ miles per hour. If the car travels $240$ miles in $4$ hours, which of the following equations can be used to determine the value of $r$ ?

4r = 240

r + 4 = 240

\frac{r}{4} = 240

r - 4 = 240

240r = 4

Show answer & explanation

Answer: $4r = 240$

Answer

The equation

4r = 240

represents the relationship because distance equals rate multiplied by time.

The relationship between distance, rate, and time is given by the formula

\text{Distance} = \text{Rate} \times \text{Time}

. Substituting the distance of

240

miles and the time of

4

hours into this formula yields

240 = r \times 4

, which is equivalent to

4r = 240

.

Step-by-Step Solution

1

Identify the formula relating distance, rate, and time.

\text{Distance} = \text{Rate} \times \text{Time}

(or

d = r \cdot t

)

This is the standard physical relationship governing constant speed motion.

2

Substitute the given values into the formula.

240 = r \cdot 4

The total distance

d

is given as

240

miles, the time

t

is

4

hours, and the speed is the variable

r

.

3

Rewrite the equation in standard algebraic format.

4r = 240

Multiplying the variable

r

by the coefficient

4

is conventionally written with the number first.

Key Concept

Translating a constant rate word problem into a one-step linear equation.

Estimated Time:45s

Question 199Question →

If $x$ is a real number such that $\sqrt{\frac{4^{x+2} + 4^{x+2} + 4^{x+2} + 4^{x+2}}{8^{x-3}}} = 64$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The value of

x

is

3

.

By rewriting the repeated addition in the numerator as

4 \times 4^{x+2} = 4^{x+3}

and converting all bases to 2, the expression inside the square root simplifies to

2^{15-x}

. Taking the square root gives

2^{\frac{15-x}{2}}

. Setting this equal to

64

(which is

2^6

) and equating the exponents yields

x = 3

.

Step-by-Step Solution

1

Rewrite the sum in the numerator

4^{x+2} + 4^{x+2} + 4^{x+2} + 4^{x+2}

as a product.

4 \times 4^{x+2} = 4^{x+3}

Adding a term to itself four times is equivalent to multiplying that term by 4.

2

Convert the base 4 numerator and base 8 denominator to base 2.

Numerator:

(2^2)^{x+3} = 2^{2x+6}

; Denominator:

(2^3)^{x-3} = 2^{3x-9}

Expressing terms with the same base allows the use of exponent laws to simplify the fraction.

3

Simplify the fraction by subtracting the denominator's exponent from the numerator's exponent.

\frac{2^{2x+6}}{2^{3x-9}} = 2^{(2x+6)-(3x-9)} = 2^{15-x}

The quotient rule for exponents states that

\frac{a^m}{a^n} = a^{m-n}

.

4

Apply the square root to the simplified fraction, set it equal to

64

, and express both sides as powers of 2.

\sqrt{2^{15-x}} = 2^{\frac{15-x}{2}} = 64 = 2^6

The square root of a term is equivalent to raising that term to the power of

\frac{1}{2}

.

5

Equate the exponents and solve for

x

.

\frac{15-x}{2} = 6 \implies 15-x = 12 \implies x = 3

If two exponential expressions with the same positive base are equal, their exponents must be equal.

Key Concept

Simplifying expressions using the laws of exponents and properties of roots

Question 200Question →

A container holds a mixture of sand, gravel, and cement. By weight, the mixture is $30\%$ sand, $\frac{2}{5}$ gravel, and the remaining portion is cement. To adjust the mixture's properties, the amount of sand is doubled, the amount of gravel is increased by $25\%$ , and the amount of cement is decreased by $50\%$ . What percentage of the new mixture, by weight, is cement?

12\%

15\%

20\%

33\frac{1}{3}\%

40\%

Show answer & explanation

Answer: $12\%$

Answer

12%

To find the correct percentage, we establish that out of an initial

100\text{ g}

of the mixture, there are

30\text{ g}

of sand,

40\text{ g}

of gravel, and

30\text{ g}

of cement. After the changes, the new weights are

60\text{ g}

of sand,

50\text{ g}

of gravel, and

15\text{ g}

of cement. The new total weight is

125\text{ g}

. Thus, cement constitutes

\frac{15}{125} = 12\%

of the new mixture.

Step-by-Step Solution

1

Determine the initial proportions and assign a hypothetical total weight of

100\text{ g}

to simplify the calculations.

Sand is

30\text{ g}

. Gravel is

\frac{2}{5} \times 100\text{ g} = 40\text{ g}

. Cement is the remaining weight:

100\text{ g} - (30\text{ g} + 40\text{ g}) = 30\text{ g}

.

Establishing concrete weights makes applying percentage changes straightforward.

2

Apply the specified weight changes to each component to find their new weights.

New sand weight:

30\text{ g} \times 2 = 60\text{ g}

. New gravel weight:

40\text{ g} \times 1.25 = 50\text{ g}

. New cement weight:

30\text{ g} \times 0.50 = 15\text{ g}

.

This calculates the individual component weights after the adjustments.

3

Calculate the new total weight of the mixture and the new percentage of cement.

New total weight:

60\text{ g} + 50\text{ g} + 15\text{ g} = 125\text{ g}

. Cement percentage:

\frac{15\text{ g}}{125\text{ g}} \times 100\% = 12\%

.

The percentage of a component in a mixture is its weight divided by the new total weight of the mixture.

Key Concept

Calculating new percentage concentrations after changes in individual component weights within a mixture.

Estimated Time:2m 0s

Pre-Algebra

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