Pre-Algebra Practice Questions — ACT — Page 8

Question 141Question →

A laboratory technician measured the concentration of active reagent in four different solutions and recorded the values in various numerical forms: Solution W contains $\frac{7}{16}$ active reagent, Solution X contains $42.5\%$ active reagent, Solution Y contains $0.44$ active reagent, and Solution Z contains $\frac{4}{9}$ active reagent. What is the correct order of the four solutions from the least concentration of active reagent to the greatest concentration?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest concentration is Solution X (

42.5\%

), Solution W (

\frac{7}{16}

), Solution Y (

0.44

), and Solution Z (

\frac{4}{9}

).

To arrange values given in different numerical formats, convert each to decimal form: Solution X equals

0.4250

, Solution W equals

0.4375

, Solution Y equals

0.4400

, and Solution Z equals

0.4444...

. Arranging these decimals from least to greatest yields Solution X, Solution W, Solution Y, then Solution Z.

Step-by-Step Solution

1

Convert all values to decimal format for direct comparison.

Solution W:

\frac{7}{16} = 0.4375

; Solution X:

42.5\% = 0.425

; Solution Y:

0.44 = 0.4400

; Solution Z:

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

Converting fractions, decimals, and percentages to a single decimal format makes order comparisons straightforward.

2

Compare the resulting decimal values place by place.

0.4250 < 0.4375 < 0.4400 < 0.4444...

Comparing digits from left to right establishes the numerical sequence.

3

Match the ordered decimals back to their corresponding solution names.

Solution X (

0.425

) < Solution W (

0.4375

) < Solution Y (

0.44

) < Solution Z (

\frac{4}{9}

)

The question requires ranking the original solution labels.

Key Concept

Converting fractions, decimals, and percentages into a uniform decimal format to compare and order rational numbers

Question 142Question →

A community library has a collection of 250 books. If $44\%$ of these books are classified as fiction, how many fiction books are in the library's collection?

Show answer & explanation

Answer: 110

Answer

There are 110 fiction books in the library's collection.

To find

44\%

of 250 books, first convert

44\%

to a decimal by dividing by 100, which is

0.44

. Then, multiply the decimal by the total number of books:

0.44 \times 250 = 110

. This represents the number of fiction books in the library.

Step-by-Step Solution

1

Convert the percentage of fiction books to a decimal.

0.44

To write a percent as a decimal, divide by 100 or move the decimal point two places to the left.

2

Multiply the decimal by the total number of books to find the number of fiction books.

110

Multiplying the decimal representing the fraction of the total by the total number of items gives the portion representing fiction books.

Key Concept

Calculating a percentage of a total amount

Estimated Time:45s

Question 143Question →

A certain metal alloy is made of copper, zinc, and nickel. By weight, $\frac{1}{4}$ of the alloy is copper, and $30\%$ of the alloy is zinc. What fraction of the alloy is nickel?

\frac{21}{20}

\frac{5}{7}

\frac{9}{20}

\frac{1}{2}

\frac{21}{40}

Show answer & explanation

Answer: $\frac{9}{20}$

Answer

The fraction of the alloy that is nickel is

\frac{9}{20}

.

The correct answer is the fraction representing the remaining part of the alloy after accounting for copper and zinc. First, convert

30\%

to the fraction

\frac{3}{10}

. The combined fraction of copper and zinc is

\frac{1}{4} + \frac{3}{10} = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}

. Subtracting this combined fraction from

1

yields the fraction of nickel:

1 - \frac{11}{20} = \frac{9}{20}

.

Step-by-Step Solution

1

Convert the percentage of zinc to a fraction.

Since

30\% = \frac{30}{100}

, this simplifies to

\frac{3}{10}

.

To add or subtract the proportions of the alloy, they must be expressed in the same form (fractions).

2

Calculate the combined fraction of copper and zinc.

\frac{1}{4} + \frac{3}{10} = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}

.

To find how much of the alloy is made of copper and zinc combined, we add their individual fractions using a common denominator of 20.

3

Subtract the combined fraction from the whole to find the fraction of nickel.

1 - \frac{11}{20} = \frac{20}{20} - \frac{11}{20} = \frac{9}{20}

.

Since the entire alloy represents a whole (

1

), the remaining portion must be nickel.

Key Concept

Expressing percentages as fractions and performing operations with fractions to solve part-to-whole problems.

Question 144Question →

What is the value of the mathematical expression below?

\frac{2^2 \times 2^3}{2} - \frac{2^4}{|-9 + 5|}

-8

12

14

20

28

Show answer & explanation

Answer: 12

Answer

12

Evaluating the first term gives 16, and evaluating the second term gives 4. Subtracting 4 from 16 results in the correct value of 12.

Step-by-Step Solution

1

Evaluate the first term:

\frac{2^2 \times 2^3}{2}

16

According to the product rule of exponents,

2^2 \times 2^3 = 2^{2+3} = 2^5 = 32

. Dividing this by the denominator yields

32 \div 2 = 16

.

2

Evaluate the second term:

\frac{2^4}{|-9 + 5|}

4

First, evaluate the expression inside the absolute value brackets:

-9 + 5 = -4

. The absolute value is

|-4| = 4

. Then evaluate the exponent in the numerator:

2^4 = 16

. Dividing the numerator by the denominator yields

16 \div 4 = 4

.

3

Subtract the second term from the first term

12

Subtracting the result of the second term from the first term yields

16 - 4 = 12

.

Key Concept

Order of Operations and Number Properties

Estimated Time:1m 30s

Question 145Question →

An explorer records the elevation, in meters relative to sea level, of four research stations: $W$ , $X$ , $Y$ , and $Z$ . Station $W$ is located at an elevation of $-15$ meters. The elevation of Station $X$ is the absolute value of the elevation of Station $W$ . The elevation of Station $Y$ is $8$ meters lower than the elevation of Station $X$ . Station $Z$ is at an elevation such that the distance between the elevations of Station $Y$ and Station $Z$ on a vertical number line is exactly $12$ meters. Which of the following could be the elevation, in meters, of Station $Z$ ?

-35

-11

-5

5

11

Show answer & explanation

Answer: $-5$

Answer

The correct elevation of Station Z could be

-5

meters.

The correct answer is

-5

meters. First, we find the elevation of Station X, which is the absolute value of Station W's elevation:

|-15| = 15

meters. Next, we determine the elevation of Station Y, which is

8

meters lower than Station X:

15 - 8 = 7

meters. Finally, we find the possible elevations for Station Z. The distance between Station Y and Station Z is

12

meters, which can be represented by the equation

|Z - 7| = 12

. This gives two possible elevations:

Z = 7 + 12 = 19

meters or

Z = 7 - 12 = -5

meters. Among the choices, only

-5

is listed.

Step-by-Step Solution

1

Find the elevation of Station X by taking the absolute value of the elevation of Station W.

The elevation of Station X is

|-15| = 15

meters.

The problem states that the elevation of Station X is the absolute value of the elevation of Station W, which is

-15

meters.

2

Calculate the elevation of Station Y by subtracting 8 meters from the elevation of Station X.

The elevation of Station Y is

15 - 8 = 7

meters.

Station Y is 8 meters lower than Station X, which is at 15 meters.

3

Set up the absolute value equation representing the distance of 12 meters between Station Y and Station Z, and solve for the two possible values of Z.

|Z - 7| = 12

, which yields

Z - 7 = 12 \implies Z = 19

or

Z - 7 = -12 \implies Z = -5

.

The distance between two points

a

and

b

on a number line is given by

|a - b|

. Since the distance between Y and Z is 12, we solve the equation to find all possible elevations.

Key Concept

Absolute value represents the distance of a number from zero on a number line, and the distance between two numbers

a

and

b

is given by

|a - b|

.

Estimated Time:2m 0s

Question 146Question →

A student took a history exam containing $80$ questions. If the student answered $15\%$ of the questions incorrectly, how many questions did the student answer correctly?

Show answer & explanation

Answer: 68

Answer

The student answered

68

questions correctly.

First, find the percentage of correct answers by subtracting the percentage of incorrect answers (

15\%

) from

100\%

, which gives

85\%

. Convert

85\%

to a decimal (

0.85

) and multiply it by the total number of questions (

80

):

80 \times 0.85 = 68

.

Step-by-Step Solution

1

Calculate the percentage of questions answered correctly.

85\%

Subtract the percentage of incorrect questions (

15\%

) from the total (

100\%

) to find the percentage of correct questions.

2

Convert the percentage of correct questions to a decimal.

0.85

Divide the percentage by

100

to express it as a decimal.

3

Multiply the total questions by the decimal value.

68

Multiplying the total number of questions by the proportion of correct answers gives the total number of correct questions:

80 \times 0.85 = 68

.

Key Concept

Calculating percentages of a whole number

Question 147Question →

On a standard number line, points $A$ , $B$ , $C$ , and $D$ have distinct integer coordinates $a$ , $b$ , $c$ , and $d$ , respectively, such that $a < b < c < d$ . The distance between $A$ and $B$ is equal to the distance between $C$ and $D$ . The distance between $B$ and $C$ is $\frac{2}{3}$ of the distance between $A$ and $B$ . If the average (arithmetic mean) of the four coordinates is $0$ and $|a - d| = 24$ , what is the coordinate of $B$ ?

Show answer & explanation

Answer: -3

Answer

The coordinate of

B

is

-3

.

The coordinate of

B

is found by setting the segment lengths to

3k

,

2k

, and

3k

based on the given ratio. Using the absolute value distance

|a - d| = 24

, we find

k = 3

, meaning the segments are

9

,

6

, and

9

. Using the coordinate sum of

0

, we get

4a + 48 = 0 \implies a = -12

. Substituting this back gives the coordinate of

B

as

-3

.

Step-by-Step Solution

1

Define segment lengths using a variable

k

based on the given ratio.

Let the distance

AB = CD = 3k

and

BC = 2k

.

This allows us to write all distances as integer multiples of a single variable since the ratio of

BC

to

AB

is

\frac{2}{3}

.

2

Set up an equation for the total distance from

A

to

D

using the absolute value

|a - d| = 24

.

Since

a < d

,

d - a = |a - d| = 24

. The sum of the segments is

3k + 2k + 3k = 8k

. Solving

8k = 24

yields

k = 3

.

The absolute value of the difference between the outermost points represents the total length of the number line segment containing all four points.

3

Express the coordinates of

b

,

c

, and

d

in terms of

a

using the calculated segment lengths.

b = a + 9

,

c = a + 15

, and

d = a + 24

.

Since the points are in order

a < b < c < d

, we add the segment lengths successively to find the coordinates.

4

Apply the average condition to solve for the coordinate

a

.

The sum of the coordinates is

4 \times 0 = 0

, so

a + (a + 9) + (a + 15) + (a + 24) = 4a + 48 = 0

, giving

a = -12

.

An average of

0

for four numbers means their sum must be

0

.

5

Find the coordinate of

B

by substituting the value of

a

into the expression for

b

.

b = -12 + 9 = -3

.

This yields the specific coordinate requested by the question.

Key Concept

Using absolute value as distance on a number line and expressing relationships between ordered coordinates algebraically.

Question 148Question →

Evaluate each of the following numerical expressions using the standard order of operations. Arrange the expressions in order of their simplified values from least to greatest.

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Answer

The correct order of the expressions from least to greatest simplified value is:

12 - 3 \times | -4 + 1 |

(value of 3),

18 \div 3 \times 2 - 2^3

(value of 4),

(6 - 8)^2 \div 2 + 3

(value of 5), and

-2^2 + 5 \times (3 - 1)

(value of 6).

Evaluating each expression according to the correct order of operations results in values of 3, 4, 5, and 6, respectively.

Step-by-Step Solution

1

Evaluate the expression

12 - 3 \times | -4 + 1 |

.

3

Simplify inside the absolute value first:

-4 + 1 = -3

. Taking the absolute value gives

|-3| = 3

. Perform multiplication before subtraction:

3 \times 3 = 9

. Subtract:

12 - 9 = 3

.

2

Evaluate the expression

18 \div 3 \times 2 - 2^3

.

4

Evaluate the exponent first:

2^3 = 8

. Perform multiplication and division from left to right:

18 \div 3 = 6

and

6 \times 2 = 12

. Subtract:

12 - 8 = 4

.

3

Evaluate the expression

(6 - 8)^2 \div 2 + 3

.

5

Simplify inside parentheses:

6 - 8 = -2

. Apply the exponent:

(-2)^2 = 4

. Divide:

4 \div 2 = 2

. Add:

2 + 3 = 5

.

4

Evaluate the expression

-2^2 + 5 \times (3 - 1)

.

6

Apply the exponent to the base 2:

-2^2 = -4

. Evaluate inside parentheses:

3 - 1 = 2

. Multiply:

5 \times 2 = 10

. Add:

-4 + 10 = 6

.

Key Concept

Using the standard order of operations (PEMDAS/GEMS) to evaluate numerical expressions containing exponents, grouping symbols, absolute values, multiplication, division, addition, and subtraction.

Question 149Question →

During a physical education class, a student ran $\frac{1}{4}$ of a mile and walked $\frac{3}{5}$ of a mile. What total distance, in miles, did the student cover by running and walking?

\frac{4}{9}

\frac{13}{20}

\frac{23}{40}

\frac{17}{20}

\frac{7}{10}

Show answer & explanation

Answer: $\frac{17}{20}$

Answer

The correct answer is

\frac{17}{20}

miles, which represents the sum of the distances covered by running and walking.

The correct answer is found by finding a common denominator for the two fractions representing the distances. The least common multiple of

4

and

5

is

20

. Converting the fractions gives

\frac{5}{20}

and

\frac{12}{20}

. Adding these values yields a total of

\frac{17}{20}

miles.

Step-by-Step Solution

1

Set up the addition expression for the two distances.

\frac{1}{4} + \frac{3}{5}

To find the total distance, the distance run and the distance walked must be added together.

2

Find the least common denominator (LCD) for the fractions.

The LCD for

4

and

5

is

20

.

Fractions can only be added when they share a common denominator.

3

Convert each fraction to an equivalent fraction with the denominator of

20

.

\frac{1 \times 5}{4 \times 5} = \frac{5}{20}

and

\frac{3 \times 4}{5 \times 4} = \frac{12}{20}

This scales the fractions correctly so they can be combined.

4

Add the numerators together while keeping the denominator the same.

\frac{5 + 12}{20} = \frac{17}{20}

Adding the numerators of fractions with like denominators gives the total fractional quantity.

Key Concept

Adding fractions with unlike denominators by finding a common denominator.

Alternative Method

Convert the fractions to decimals before adding.

\frac{1}{4}

is equivalent to

0.25

, and

\frac{3}{5}

is equivalent to

0.60

. Adding these decimals yields

0.25 + 0.60 = 0.85

. Converting the decimal back to a fraction results in

\frac{85}{100} = \frac{17}{20}

.

Estimated Time:45s

Question 150Question →

A laboratory mixture is prepared by combining three liquid solutions: Solution X, Solution Y, and Solution Z. Initially, Solution X makes up $20\%$ of the total volume of the mixture. Solution Y's volume is $\frac{1}{3}$ of the combined volume of Solution X and Solution Z, and Solution Z makes up the remaining portion of the mixture. If $3.0$ liters of Solution Z are added to the mixture, Solution Z then constitutes exactly $58\%$ of the new total volume. What was the original total volume, in liters, of the mixture before Solution Z was added?

15

24

35

42

100

Show answer & explanation

Answer: 42

Answer

The original total volume of the mixture was 42 liters.

The correct option represents the original volume of

42

liters. Setting the original total volume to

V

, we find that Solution X is

0.20V

. Since Solution Y is one-third of the combined volume of X and Z, we write

Y = \frac{1}{3}(V - Y)

, which simplifies to

Y = 0.25V

. The remaining portion, Solution Z, must be

V - 0.20V - 0.25V = 0.55V

. Adding

3.0

liters of Solution Z increases both the volume of Solution Z and the total volume of the mixture, yielding the proportion

(0.55V + 3) / (V + 3) = 0.58

. Solving this linear equation results in

0.03V = 1.26

, which gives

V = 42

.

Step-by-Step Solution

1

Express the initial fraction of Solution X as a decimal.

Solution X makes up

0.20

of the total volume.

Converting

20\%

to a decimal simplifies subsequent algebraic modeling.

2

Set up an equation for the fraction of Solution Y in terms of the total volume

V

.

Solution Y makes up

0.25

of the total volume.

If Solution Y's volume is

\frac{1}{3}

of the combined volume of Solution X and Solution Z, then

Y = \frac{1}{3}(V - Y)

. Multiplying by

3

gives

3Y = V - Y

, which simplifies to

4Y = V

, or

Y = 0.25V

.

3

Determine the initial fraction of Solution Z in the mixture.

Solution Z initially makes up

0.55

of the total volume.

Subtracting the fractions of Solution X and Solution Y from the whole yields

1 - 0.20 - 0.25 = 0.55

.

4

Set up and solve the equation representing the addition of Solution Z.

V = 42

liters

Adding

3.0

liters of Solution Z increases the volume of Solution Z to

0.55V + 3

and the total volume to

V + 3

. The equation is

\frac{0.55V + 3}{V + 3} = 0.58

. Multiplying both sides by

V + 3

yields

0.55V + 3 = 0.58(V + 3) \implies 0.55V + 3 = 0.58V + 1.74

. Rearranging terms gives

3 - 1.74 = 0.58V - 0.55V \implies 1.26 = 0.03V \implies V = 42

.

Key Concept

Solving multi-step mixture word problems using equations involving fractions, decimals, and percentages.

Alternative Method

Instead of variables for the total volume, solve the problem by tracking parts. Initially, the mixture consists of

20\%

Solution X,

25\%

Solution Y, and

55\%

Solution Z. The ratio of the volume of Solution Z to the combined volume of Solution X and Solution Y is

55 : 45

, which simplifies to

11 : 9

. Let the volume of X and Y combined be

9x

liters, and the initial volume of Z be

11x

liters, making the initial total volume

20x

liters. Adding

3.0

liters of Solution Z does not change the combined volume of X and Y (

9x

liters). In the new mixture, Solution Z constitutes

58\%

, meaning the combined volume of X and Y must constitute

100\% - 58\% = 42\%

of the new mixture. Thus, the new total volume is

9x / 0.42 = 150x / 7

liters. The difference between the new and original total volumes is the

3.0

liters added:

(150x / 7) - 20x = 3 \implies 10x / 7 = 3 \implies x = 2.1

. The original total volume was

20x = 20(2.1) = 42

liters.

Estimated Time:3m 0s

Question 151Question →

A landscaping company uses a grass seed mixture containing perennial ryegrass and Kentucky bluegrass in a ratio of $5:3$ by weight. If a bag of this mixture contains $15$ pounds of Kentucky bluegrass, how many pounds of perennial ryegrass does it contain?

9

24

25

40

75

Show answer & explanation

Answer: 25

Answer

25

The ratio of perennial ryegrass to Kentucky bluegrass is

5:3

, which means for every

5

pounds of ryegrass, there are

3

pounds of bluegrass. Setting up the proportion

\frac{5}{3} = \frac{x}{15}

and solving for

x

gives

3x = 75

, which simplifies to

x = 25

pounds of perennial ryegrass.

Step-by-Step Solution

1

Set up a proportion using the ratio of perennial ryegrass to Kentucky bluegrass.

\frac{5}{3} = \frac{x}{15}

, where

x

represents the weight of perennial ryegrass.

The ratio of perennial ryegrass to Kentucky bluegrass by weight is

5:3

, and the actual weight of Kentucky bluegrass is

15

pounds.

2

Solve the proportion for

x

by cross-multiplying.

3x = 5 \times 15 \implies 3x = 75

Cross-multiplication allows us to clear the denominators to solve for the unknown value.

3

Divide both sides of the equation by

3

.

x = 25

Isolating

x

gives the weight of perennial ryegrass in pounds.

Key Concept

Solving proportions based on a given part-to-part ratio

Alternative Method

Identify the scaling factor: since the ratio of Kentucky bluegrass is

3

parts and the actual weight is

15

pounds, each part of the ratio represents

15 \div 3 = 5

pounds. Multiplying the

5

parts of perennial ryegrass by this factor of

5

gives

5 \times 5 = 25

pounds.

Estimated Time:45s

Question 152Question →

A positive integer $n$ is a multiple of $7$ , and the greatest common factor of $n$ and $120$ is $15$ . Which of the following is a possible value for $n$ ?

75

105

120

210

420

Show answer & explanation

Answer: 105

Answer

105

The correct answer is 105. Since the greatest common factor of

n

and 120 is 15,

n

must be a multiple of 15. We are also given that

n

is a multiple of 7. Since 7 and 15 share no common factors other than 1,

n

must be a multiple of

7 \times 15 = 105

. Let

n = 105k

for some positive integer

k

. We must ensure that the greatest common factor of

105k

and 120 is exactly 15. Since

\text{GCF}(105k, 120) = 15 \times \text{GCF}(7k, 8)

, we require

\text{GCF}(7k, 8) = 1

, which means

k

must be an odd integer. Choosing the smallest positive odd integer

k = 1

yields

n = 105

, which is a possible value.

Step-by-Step Solution

1

Analyze the prime factorization of 120 and the greatest common factor requirement.

Since the greatest common factor (

\text{GCF}

) of

n

and 120 is 15,

n

must be divisible by 15. The prime factorization of 120 is

2^3 \times 3 \times 5

, and

15 = 3 \times 5

. Therefore,

n

must contain the prime factors 3 and 5, but cannot contain any powers of 2.

To establish which prime factors

n

must and must not have to satisfy the GCF condition.

2

Combine the GCF divisibility condition with the multiple of 7 constraint.

Since

n

is a multiple of 7 and also a multiple of 15, and because 7 and 15 are coprime (they share no common prime factors),

n

must be a multiple of

7 \times 15 = 105

. Let

n = 105k

for some positive integer

k

.

To find the general form of

n

by finding the least common multiple of the required factor components.

3

Test the constraints on

k

to identify valid values for

n

.

We need

\text{GCF}(105k, 120) = 15

. This can be written as

\text{GCF}(15 \times 7k, 15 \times 8) = 15 \times \text{GCF}(7k, 8) = 15

, which requires

\text{GCF}(7k, 8) = 1

. Since 7 and 8 are coprime,

k

must not share any common factors with 8 (i.e.,

k

must be odd). The smallest positive odd integer is

k = 1

, which gives

n = 105

.

To determine the valid values of

n

and check them against the options.

Key Concept

Using prime factorizations to determine greatest common factors under multiple divisibility constraints.

Estimated Time:1m 0s

Question 153Question →

A positive integer $K$ is a multiple of $10$ and has exactly $8$ positive factors (including $1$ and $K$ ). What is the least possible value of $K$ ?

Show answer & explanation

Answer: 30

Answer

The least possible value of

K

is

30

.

The correct answer is

30

because it is the smallest multiple of

10

that has exactly

8

positive factors (

1, 2, 3, 5, 6, 10, 15,

and

30

).

Step-by-Step Solution

1

Determine the prime factors required for a multiple of

10

.

K

must have at least one factor of

2

and at least one factor of

5

.

Since

10 = 2 \times 5

, any multiple of

10

must be divisible by both

2

and

5

.

2

Set up the factor counting formula.

The number of factors is

(a+1)(b+1)(c+1)\cdots = 8

, where

a, b, c, \dots

are the exponents of the prime factorization.

The number of positive divisors of

p_1^a p_2^b \cdots

is found by adding

1

to each exponent and multiplying the results.

3

Evaluate the smallest multiples of

10

to find the first one with exactly

8

factors.

10

has

4

factors (

1, 2, 5, 10

);

20

has

6

factors (

1, 2, 4, 5, 10, 20

);

30

has

8

factors (

1, 2, 3, 5, 6, 10, 15, 30

).

Testing the positive multiples of

10

in ascending order ensures that the first number with exactly

8

factors is the smallest possible value.

Key Concept

Factors, Multiples, and Prime Factorization

Question 154Question →

Arrange the following numbers in order from least to greatest: $\frac{2}{5}$ , $0.38$ , $45\%$ , and $0.42$ .

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the values from least to greatest is

0.38

,

\frac{2}{5}

,

0.42

, and

45\%

.

Converting all numbers to decimals allows for direct comparison:

0.38

remains

0.38

; the fraction

\frac{2}{5}

equals

0.40

;

0.42

remains

0.42

; and

45\%

equals

0.45

. Comparing these decimal values, we get

0.38 < 0.40 < 0.42 < 0.45

. Therefore, the correct order from least to greatest is

0.38

,

\frac{2}{5}

,

0.42

, and

45\%

.

Step-by-Step Solution

1

Convert the fraction to a decimal.

\frac{2}{5} = 0.40

Converting all values to decimals makes them easier to compare.

2

Convert the percentage to a decimal.

45\% = 0.45

Percentages represent values out of 100, so

45\% = \frac{45}{100} = 0.45

.

3

Compare the four decimal numbers.

0.38 < 0.40 < 0.42 < 0.45

By comparing the tenths and hundredths place, we find the correct order from smallest to largest.

Key Concept

Comparing and ordering fractions, decimals, and percents by converting them to a common decimal format.

Question 155Question →

A scientist monitors the temperature in three different chambers, $A$ , $B$ , and $C$ . The temperature in Chamber $B$ is $T^\circ\text{C}$ . The temperature in Chamber $A$ is always $5^\circ\text{C}$ colder than the temperature in Chamber $B$ , and the temperature in Chamber $C$ is always $9^\circ\text{C}$ warmer than the temperature in Chamber $B$ . If $T$ must be an integer, and the sum of the absolute values of the temperatures in all three chambers is minimized, what is the value of $T$ ?

-9

-1

0

2

5

Show answer & explanation

Answer: 0

Answer

The correct answer is

0

.

The correct answer is the value

0

. The sum of the absolute values of the temperatures in all three chambers is represented by the function

S(T) = |T - 5| + |T| + |T + 9|

. Geometrically, this sum represents the total distance from a point

T

to the coordinates

5

,

0

, and

-9

on a number line. For any set of points, the sum of absolute differences is minimized at their median. The median of the set

\{-9, 0, 5\}

is

0

. Evaluating the sum at

T = 0

yields

|0 - 5| + |0| + |0 + 9| = 5 + 0 + 9 = 14

. Any other integer choice yields a larger sum (for example, a value of

1

or

-1

yields a sum of

15

).

Step-by-Step Solution

1

Write expressions for the temperatures of the three chambers in terms of

T

.

The temperature in Chamber

A

is

T - 5

, in Chamber

B

is

T

, and in Chamber

C

is

T + 9

.

To represent the temperature of each chamber relative to Chamber

B

.

2

Write the sum of the absolute values of the three temperatures.

The sum is represented by the expression

S(T) = |T - 5| + |T| + |T + 9|

.

To formulate the function that needs to be minimized.

3

Find the key values that make each absolute value term zero.

The points are

5

,

0

, and

-9

. Ordered on a number line, these points are

-9

,

0

, and

5

.

The sum of absolute differences is geometrically equivalent to the sum of distances on a number line, which is minimized at the median of the points.

4

Evaluate the sum of absolute values at the median value,

T = 0

, and compare it to other values.

For

T = 0

,

S(0) = |-5| + |0| + |9| = 14

. For

T = 1

,

S(1) = |-4| + |1| + |10| = 15

. For

T = -1

,

S(-1) = |-6| + |-1| + |8| = 15

. Thus, the minimum sum occurs at

T = 0

.

To verify that the median yields the minimum sum of absolute values.

Key Concept

Minimizing the sum of absolute values of linear terms by finding the median of their zero-points on a number line.

Question 156Question →

A water purification plant processes raw water through three successive filtration stages: Stage A, Stage B, and Stage C. Stage A removes $\frac{3}{8}$ of the impurities present in the raw water. Stage B removes $40\%$ of the remaining impurities. Stage C removes $0.75$ of the impurities that remain after Stage B. If $4.5$ kilograms of impurities are successfully removed in Stage C, how many kilograms of impurities were originally in the raw water?

Show answer & explanation

Answer: 16

Answer

The original amount of impurities in the raw water was

16

kilograms.

The correct answer is

16

kg. To find this, we express the remaining impurities at each stage as a fraction of the initial amount

x

. After Stage A,

\frac{5}{8}x

remains. In Stage B,

40\%

is removed, meaning

60\%

of

\frac{5}{8}x

, or

\frac{3}{8}x

, remains. In Stage C,

0.75

(or

\frac{3}{4}

) of this remainder is removed, which is

\frac{3}{4} \times \frac{3}{8}x = \frac{9}{32}x

. Setting

\frac{9}{32}x = 4.5

gives

x = 16

.

Step-by-Step Solution

1

Represent the initial mass of impurities in the raw water with a variable.

Let

x

be the initial kilograms of impurities.

Establishing a variable allows setting up an equation to track the changes through each stage.

2

Calculate the remaining impurities after Stage A.

Impurities remaining =

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

kg.

Stage A removes

\frac{3}{8}

of the initial impurities, so

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

of the impurities remain.

3

Calculate the impurities removed and remaining after Stage B.

Impurities removed in Stage B =

0.40 \times \frac{5}{8}x = \frac{1}{4}x

kg. Impurities remaining after Stage B = \frac{5}{8}x - \frac{1}{4}x = \frac{3}{8}x$ kg.

Stage B removes

40\%

of the impurities that remained after Stage A. Subtracting the removed portion from the starting amount for this stage yields the remaining fraction.

4

Express the impurities removed in Stage C.

Impurities removed in Stage C =

0.75 \times \frac{3}{8}x = \frac{9}{32}x

kg.

Stage C removes

0.75

(or

\frac{3}{4}

) of the impurities remaining after Stage B.

5

Equate the Stage C expression to the given value and solve for

x

.

\frac{9}{32}x = 4.5 \implies x = 4.5 \times \frac{32}{9} = 16

kg.

The problem states that

4.5

kg of impurities are removed in Stage C, so solving this equation yields the initial value.

Key Concept

Solving multi-step word problems involving successive applications of fractions, decimals, and percentages.

Alternative Method

Working backwards from the final stage can simplify the calculations. Since Stage C removes

0.75

of the impurities remaining after Stage B, and this amount equals

4.5

kg, the amount remaining after Stage B is

\frac{4.5}{0.75} = 6

kg. Since Stage B removes

40\%

of the impurities remaining after Stage A, the

6

kg represents

100\% - 40\% = 60\%

of the impurities remaining after Stage A. Thus, the amount remaining after Stage A is

\frac{6}{0.60} = 10

kg. Finally, since Stage A removes

\frac{3}{8}

of the original impurities, the

10

kg represents

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

of the original impurities. The original amount is therefore

10 \times \frac{8}{5} = 16

kg.

Estimated Time:3m 0s

Question 157Question →

Two integers, $x$ and $y$ , are positioned on a number line. The distance between $x$ and $1$ is twice the distance between $y$ and $2$ . If the distance between $x$ and $y$ is exactly $5$ units, what is the sum of all possible values of $x + y$ ?

Show answer & explanation

Answer: 17

Answer

The sum of all possible values of

x + y

is

17

.

Representing the distances algebraically yields the system

|x - 1| = 2|y - 2|

and

|x - y| = 5

. Since

x

and

y

must be integers, splitting these equations into positive and negative cases yields exactly three valid integer coordinate pairs:

(13, 8)

,

(5, 0)

, and

(-7, -2)

. The sums (

x + y

) for these pairs are

21

,

5

, and

-9

, respectively. Adding these possible sums together gives a final total of

17

.

Step-by-Step Solution

1

Set up the absolute value expressions representing the distances.

|x - 1| = 2|y - 2|

and

|x - y| = 5

Distance on a number line between two points

a

and

b

is mathematically defined as

|a - b|

.

2

Split the distance condition

|x - y| = 5

into two coordinate cases.

x = y + 5

or

x = y - 5

An absolute value equation of the form

|A| = B

splits into

A = B

or

A = -B

.

3

Substitute

x = y + 5

into the first equation and solve for integer values of

y

.

y = 8

(which gives

x = 13

) and

y = 0

(which gives

x = 5

)

This generates the first set of valid integer coordinates satisfying all constraints.

4

Substitute

x = y - 5

into the first equation and solve for integer values of

y

.

y = -2

(which gives

x = -7

); the second algebraic option

y = 10/3

is discarded because it is not an integer

This generates the remaining valid integer coordinates satisfying all constraints.

5

Sum the value of

x + y

for all three valid coordinate pairs.

(13 + 8) + (5 + 0) + (-7 - 2) = 21 + 5 - 9 = 17

The question asks for the sum of all possible values of the expression

x + y

.

Key Concept

Using absolute value to represent distances on a number line and solving systems of absolute value equations under integer constraints.

Estimated Time:2m 30s

Question 158Question →

In a school election for student council president, the ratio of the number of votes received by Candidate A to the number of votes received by Candidate B was $2:3$ . If Candidate B received $120$ votes, how many votes did Candidate A receive?

48

72

80

180

200

Show answer & explanation

Answer: 80

Answer

Candidate A received 80 votes.

The ratio of the number of votes for Candidate A to Candidate B is

2:3

. Since Candidate B received

120

votes, we can set up the proportion

\frac{\text{Votes for A}}{120} = \frac{2}{3}

. Multiplying both sides by

120

gives

\text{Votes for A} = 120 \times \frac{2}{3} = 80

.

Step-by-Step Solution

1

Identify the given ratio and values.

The ratio of Candidate A's votes to Candidate B's votes is

2:3

, and Candidate B received

120

votes.

This sets up the proportion relationship where the ratio of Candidate A's votes to Candidate B's votes is equal to 2 to 3.

2

Set up a proportion to solve for Candidate A's votes.

\frac{x}{120} = \frac{2}{3}

, where

x

represents the number of votes Candidate A received.

This equates the simplified ratio of votes to the actual number of votes.

3

Solve for

x

by multiplying both sides by

120

.

x = 120 \times \frac{2}{3} = 80

.

This isolates the variable

x

to find the correct number of votes Candidate A received.

Key Concept

Solving proportions and scaling ratios based on a given part.

Estimated Time:45s

Question 159Question →

A chemical mixture is kept in a temperature-controlled chamber where the temperature $T$ , in degrees Celsius, is restricted to the range $-10 \le T \le 10$ . The mixture remains stable as long as the temperature satisfies the inequality:

2 - 3|T - 2| \ge -10

What is the sum of all integer values of $T$ in this range for which the mixture is stable?

19

-14

18

-18

20

Show answer & explanation

Answer: 18

Answer

18

The correct answer is 18. Subtracting 2 from both sides of the inequality

2 - 3|T - 2| \ge -10

gives

-3|T - 2| \ge -12

. Dividing by

-3

and reversing the inequality sign yields

|T - 2| \le 4

, which expands to the compound inequality

-4 \le T - 2 \le 4

. Adding 2 to all parts gives

-2 \le T \le 6

. All integers in this range (

-2, -1, 0, 1, 2, 3, 4, 5, 6

) are within the chamber's allowed range of

[-10, 10]

. Their sum is

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 18

.

Step-by-Step Solution

1

Isolate the absolute value expression by subtracting 2 from both sides.

-3|T - 2| \ge -12

To solve an absolute value inequality, we must first isolate the absolute value term on one side.

2

Divide both sides by

-3

and reverse the inequality sign.

|T - 2| \le 4

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

3

Rewrite the absolute value inequality as a compound inequality.

-4 \le T - 2 \le 4

An inequality of the form

|x| \le a

(where

a \ge 0

) is equivalent to

-a \le x \le a

.

4

Solve for

T

by adding 2 to all three parts of the inequality.

-2 \le T \le 6

Adding 2 isolates the variable

T

to find the range of stable temperatures.

5

Identify all integer values of

T

in the stable range and calculate their sum.

Sum = 18

The integers in the interval

[-2, 6]

are

-2, -1, 0, 1, 2, 3, 4, 5, 6

. Since the chamber is restricted to

[-10, 10]

, all of these values are valid. Summing them:

(-2) + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 = 18

.

Key Concept

Solving absolute value inequalities and performing operations with signed integers.

Question 160Question →

A baker is preparing a recipe that calls for $2$ times the sum of $\frac{1}{4}$ cup of milk and $\frac{2}{5}$ cup of water. What is the total volume, in cups, of these two liquid ingredients combined?

\frac{2}{3}

\frac{9}{10}

\frac{13}{10}

\frac{34}{35}

\frac{13}{20}

Show answer & explanation

Answer: $\frac{13}{10}$

Answer

The correct answer is

\frac{13}{10}

cups.

To find the total volume, first determine the sum of the two fractions by finding a common denominator:

\frac{1}{4} + \frac{2}{5} = \frac{5}{20} + \frac{8}{20} = \frac{13}{20}

cups. Then, multiply this sum by

2

as required by the recipe:

2 \times \frac{13}{20} = \frac{26}{20} = \frac{13}{10}

cups.

Step-by-Step Solution

1

Find a common denominator for the two fractions to be added.

The common denominator for 4 and 5 is 20, converting the fractions to

\frac{5}{20}

and

\frac{8}{20}

.

Fractions must have the same denominator before they can be added.

2

Add the two converted fractions.

\frac{5}{20} + \frac{8}{20} = \frac{13}{20}

.

To find the sum of the ingredients inside the parentheses.

3

Multiply the sum of the fractions by 2.

2 \times \frac{13}{20} = \frac{26}{20} = \frac{13}{10}

.

The recipe calls for 2 times the sum of the two liquid ingredients.

Key Concept

Adding fractions with unlike denominators and performing operations in the correct order.

Pre-Algebra

Answer

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Common Mistakes

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