All practice questions

Arrange the following fractions and mixed numbers in ascending order (from smallest to largest).

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Answer

The correct ascending order is

\frac{7}{8}

,

1 \frac{1}{4}

,

\frac{3}{2}

, and

1 \frac{3}{4}

.

The proper fraction

\frac{7}{8}

is the only value less than 1, placing it first. Converting

\frac{3}{2}

to

1 \frac{1}{2}

(or

1.5

) allows easy comparison with

1 \frac{1}{4}

(1.25) and

1 \frac{3}{4}

(1.75), confirming the correct ascending sequence.

Step-by-Step Solution

1

Identify values less than 1.

\frac{7}{8}

is the only proper fraction (less than 1), making it the smallest.

Proper fractions are always less than mixed numbers or improper fractions.

2

Convert the improper fraction to a mixed number to make comparison easier.

\frac{3}{2} = 1 \frac{1}{2}

.

Matching the format of the other numbers allows for a direct comparison of the fractional parts.

3

Find a common denominator for the fractional parts of the values greater than 1 (

1 \frac{1}{4}

,

1 \frac{1}{2}

, and

1 \frac{3}{4}

).

The common denominator is 4.

1 \frac{1}{2}

becomes

1 \frac{2}{4}

.

Common denominators allow you to directly compare the numerators.

4

Order the mixed numbers by their fractional parts.

1 \frac{1}{4}

is smaller than

1 \frac{2}{4}

(which is

\frac{3}{2}

), and

1 \frac{2}{4}

is smaller than

1 \frac{3}{4}

.

Since the whole numbers are identical, the fractions determine the order.

Key Concept

Ordering mixed numbers and improper fractions

Question 382Question →

Evaluate the following mathematical expressions involving decimals. What is the correct sequence of these expressions when ordered from least to greatest value?

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Answer

The correct order from least to greatest is:

0.15 \times 40 - 14.4 \div 1.2

, followed by

10.5 - 3.2 \times 2.5

, then

(4.2 - 1.8) \div 0.6 + 0.5

, and finally

2.4 \times 0.5 + 1.8 \div 0.3

.

Evaluating each expression using the correct order of operations yields final decimal values of

7.2

,

2.5

,

-6.0

, and

4.5

. Arranging these values numerically from least to greatest provides the correct structural sequence.

Step-by-Step Solution

1

Evaluate the first expression:

2.4 \times 0.5 + 1.8 \div 0.3

.

1.2 + 6 = 7.2

According to the order of operations, multiplication and division are performed left-to-right before addition.

2

Evaluate the second expression:

10.5 - 3.2 \times 2.5

.

10.5 - 8.0 = 2.5

Multiplication must be performed before subtraction.

3

Evaluate the third expression:

0.15 \times 40 - 14.4 \div 1.2

.

6.0 - 12 = -6.0

Perform multiplication and division first. Then subtract the larger value from the smaller value, resulting in a negative number.

4

Evaluate the fourth expression:

(4.2 - 1.8) \div 0.6 + 0.5

.

2.4 \div 0.6 + 0.5 = 4 + 0.5 = 4.5

Operations inside parentheses take highest priority, followed by division, and finally addition.

5

Order the evaluated results from least to greatest.

-6.0 < 2.5 < 4.5 < 7.2

Arranging the numerical values based on their algebraic magnitude provides the final sequence.

Key Concept

Order of Operations with Decimals

Alternative Method

Convert all terminating decimals to fractions (e.g.,

0.5 = \frac{1}{2}

,

0.3 = \frac{3}{10}

) before performing the operations to avoid decimal placement and alignment errors.

Estimated Time:1m 30s

Question 383Question →

Arrange the following pairs of integers in ascending order based on the value of their Greatest Common Factor (GCF).

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Answer

The correct ascending order based on the Greatest Common Factor (GCF) is:

32

and

40

(GCF = 8),

36

and

48

(GCF = 12),

42

and

56

(GCF = 14),

45

and

60

(GCF = 15).

By evaluating the Greatest Common Factor for each pair, we find that the GCF of

32

and

40

is

8

; the GCF of

36

and

48

is

12

; the GCF of

42

and

56

is

14

; and the GCF of

45

and

60

is

15

. The correct ascending order is

8

,

12

,

14

,

15

.

Step-by-Step Solution

1

Find the GCF for

45

and

60

.

The GCF is

15

.

The factors of

45

are

1, 3, 5, 9, 15, 45

. The factors of

60

include

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

. The largest shared factor is

15

.

2

Find the GCF for

32

and

40

.

The GCF is

8

.

The largest number that divides exactly into both

32

and

40

is

8

.

3

Find the GCF for

42

and

56

.

The GCF is

14

.

The largest number that divides exactly into both

42

and

56

is

14

(

42 = 14 \times 3

,

56 = 14 \times 4

).

4

Find the GCF for

36

and

48

.

The GCF is

12

.

The largest number that divides exactly into both

36

and

48

is

12

.

5

Order the pairs based on the calculated GCFs.

The sequence from smallest to largest GCF is

8

,

12

,

14

,

15

.

Arranging the values in ascending order determines the correct sequence.

Key Concept

Calculating the Greatest Common Factor (GCF) of pairs of integers.

Alternative Method

You can also use prime factorization for each number to find the GCF by multiplying the lowest powers of all common prime factors between the pairs.

Estimated Time:1m 30s

Question 384Question →

Calculate the value of each decimal expression below. Arrange the results in ascending order (from smallest to largest).

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Answer

The correct order from smallest to largest is:

1.5 \times 2

, followed by

1.2 + 2.4

, then

8.4 \div 2

, and finally

5.8 - 1.3

.

Evaluating each mathematical expression individually yields 4.5, 3.6, 4.2, and 3.0 respectively. Arranging these numerical results in ascending order gives 3.0, 3.6, 4.2, and 4.5. This directly maps to the expressions:

1.5 \times 2

, then

1.2 + 2.4

, then

8.4 \div 2

, and finally

5.8 - 1.3

.

Step-by-Step Solution

1

Calculate the value of the first expression.

5.8 - 1.3 = 4.5

To determine the numerical value for comparison.

2

Calculate the value of the second expression.

1.2 + 2.4 = 3.6

To determine the numerical value for comparison.

3

Calculate the value of the third expression.

8.4 \div 2 = 4.2

To determine the numerical value for comparison.

4

Calculate the value of the fourth expression.

1.5 \times 2 = 3.0

To determine the numerical value for comparison.

5

Compare the four calculated values and order them from smallest to largest.

3.0 < 3.6 < 4.2 < 4.5

To establish the final ascending sequence required by the question.

Key Concept

Performing basic operations (addition, subtraction, multiplication, division) with decimals and comparing their magnitudes.

Estimated Time:45s

Question 385Question →

A communications technician is configuring signal transponders. The synchronization cycle for each transponder is determined by the Least Common Multiple (LCM) of its two input frequencies (measured in Hz). Arrange the following pairs of input frequencies in ascending order based on their resulting synchronization cycle (smallest LCM to largest LCM).

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Answer

The correct order from smallest to largest LCM is: 12 Hz and 18 Hz (LCM=36), 16 Hz and 24 Hz (LCM=48), 10 Hz and 25 Hz (LCM=50), 8 Hz and 14 Hz (LCM=56), followed by 15 Hz and 20 Hz (LCM=60).

The correct sequence is determined by finding the LCM of each frequency pair and listing them in ascending order. The calculated LCMs are 36, 48, 50, 56, and 60 respectively.

Step-by-Step Solution

1

Calculate the Least Common Multiple (LCM) for the first pair,

12\text{ Hz}

and

18\text{ Hz}

.

The LCM of

12

and

18

is

36

.

The multiples of

18

are

18, 36, 54

. Since

36

is divisible by

12

(

12 \times 3 = 36

), it is the lowest common multiple.

2

Calculate the LCM for

16\text{ Hz}

and

24\text{ Hz}

.

The LCM of

16

and

24

is

48

.

The multiples of

24

are

24, 48

. Since

48

is divisible by

16

(

16 \times 3 = 48

), it is the lowest common multiple.

3

Calculate the LCM for

10\text{ Hz}

and

25\text{ Hz}

.

The LCM of

10

and

25

is

50

.

The multiples of

25

are

25, 50

. Since

50

is divisible by

10

(

10 \times 5 = 50

), it is the lowest common multiple.

4

Calculate the LCM for

8\text{ Hz}

and

14\text{ Hz}

.

The LCM of

8

and

14

is

56

.

Using prime factorizations:

8 = 2^3

and

14 = 2 \times 7

. The LCM takes the highest powers:

2^3 \times 7 = 8 \times 7 = 56

.

5

Calculate the LCM for

15\text{ Hz}

and

20\text{ Hz}

.

The LCM of

15

and

20

is

60

.

Using prime factorizations:

15 = 3 \times 5

and

20 = 2^2 \times 5

. The LCM takes the highest powers:

2^2 \times 3 \times 5 = 60

.

6

Arrange the pairs in ascending order based on their calculated LCMs.

36 < 48 < 50 < 56 < 60

The stem requires ordering the frequencies from the smallest resulting synchronization cycle (smallest LCM) to the largest.

Key Concept

Calculating the Least Common Multiple (LCM) to synchronize multiple periodic events.

Question 386Question →

Determine the final value of each arithmetic expression below. What is the correct sequence when the expressions are arranged from smallest to largest result?

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Answer

The correct sequence from smallest to largest is:

0.18 + 0.42

,

3.5 \div 5

,

2.1 - 1.3

, and

0.15 \times 6

.

To arrange the sequence properly, first calculate their values:

0.18 + 0.42 = 0.6

,

3.5 \div 5 = 0.7

,

2.1 - 1.3 = 0.8

, and

0.15 \times 6 = 0.9

. By comparing these results side by side, the accurate ascending order is

0.6

,

0.7

,

0.8

, and

0.9

.

Step-by-Step Solution

1

Calculate the sum for the fourth expression.

0.18 + 0.42 = 0.60

, which is

0.6

.

Adding the hundredths and tenths places properly yields

0.6

.

2

Calculate the quotient for the third expression.

3.5 \div 5 = 0.7

.

Since

35 \div 5 = 7

, dividing

3.5

by

5

gives

0.7

.

3

Calculate the difference for the first expression.

2.1 - 1.3 = 0.8

.

Borrowing from the ones place allows subtracting

1.3

from

2.1

to get

0.8

.

4

Calculate the product for the second expression.

0.15 \times 6 = 0.90

, which is

0.9

.

Multiplying

15

by

6

gives

90

. Since the original factors have a total of two decimal places, the result is

0.90

.

5

Order the evaluated results from smallest to largest.

0.6 < 0.7 < 0.8 < 0.9

.

Comparing the tenths digit of each calculated value provides the final arrangement.

Key Concept

Performing basic arithmetic operations with decimals and comparing their magnitudes.

Question 387Question →

A biomedical researcher is cataloging the diameters of four microscopic specimens. To properly calibrate the imaging equipment, arrange the specimens in descending order based on their diameter, from the largest to the smallest.

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Answer

The correct descending order from largest to smallest is Specimen Beta, Specimen Delta, Specimen Gamma, and Specimen Alpha.

By converting all measurements into standard scientific notation (

a \times 10^n

), their magnitudes can be compared directly. Specimen Beta has the largest exponent (

-5

). The remaining three specimens all reduce to an exponent of

-6

. Ordering those three by their coefficients (

9.0

,

8.5

, and

6.5

) yields the final sequence.

Step-by-Step Solution

1

Convert Specimen Beta from standard decimal form to scientific notation.

0.000072

meters becomes

7.2 \times 10^{-5}

meters.

A uniform mathematical format is necessary to accurately compare the magnitudes of all specimens.

2

Normalize the coefficient for Specimen Gamma.

850 \times 10^{-8}

becomes

8.5 \times 10^{-6}

.

Proper scientific notation requires the coefficient to be greater than or equal to 1 and less than 10. Moving the decimal two places left increases the exponent by 2.

3

Normalize the coefficient for Specimen Delta.

0.9 \times 10^{-5}

becomes

9.0 \times 10^{-6}

.

Moving the decimal one place right to create a valid coefficient decreases the exponent by 1.

4

Compare the exponents first, then the coefficients to determine the descending order.

7.2 \times 10^{-5} > 9.0 \times 10^{-6} > 8.5 \times 10^{-6} > 6.5 \times 10^{-6}

.

A larger exponent (

-5

vs

-6

) dictates a larger overall number. For values sharing the identical

-6

exponent, the descending order is determined directly by their coefficients (

9.0 > 8.5 > 6.5

).

Key Concept

Normalizing Coefficients and Comparing Magnitudes in Scientific Notation

Estimated Time:1m 30s

Question 388Question →

A logistics specialist is calculating the weight of four different cargo pallets in kilograms. Based on the provided mathematical expressions for each pallet, arrange them in order from lightest to heaviest.

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Answer

The correct order from lightest to heaviest is Pallet S (

47.95

kg), Pallet Q (

48.1

kg), Pallet P (

48.2

kg), and Pallet R (

48.4

kg).

The correct sequence is determined by first applying the proper order of operations (PEMDAS) to compute the exact weight of each pallet. Multiplying and dividing before adding and subtracting yields the values: Pallet S is

47.95

kg, Pallet Q is

48.1

kg, Pallet P is

48.2

kg, and Pallet R is

48.4

kg. Comparing these final decimal figures starting from the greatest place value provides the correct ascending order.

Step-by-Step Solution

1

Calculate the exact weight of Pallet S.

9.5 \times 5.4 - 3.35 = 51.3 - 3.35 = 47.95

Applying the order of operations, multiplication is performed before subtraction.

2

Calculate the exact weight of Pallet Q.

12.5 + 7.12 \times 5 = 12.5 + 35.6 = 48.1

Multiplication precedes addition. Aligning the decimals properly ensures correct addition.

3

Calculate the exact weight of Pallet P.

60.2 - 4.8 \times 2.5 = 60.2 - 12 = 48.2

Multiply first to get exactly

12

, then subtract from

60.2

.

4

Calculate the exact weight of Pallet R.

58.08 \div 1.2 = 580.8 \div 12 = 48.4

Shift the decimal point one place to the right in both the dividend and divisor to compute the division.

5

Compare the final decimal values to determine the ascending order.

47.95 < 48.1 < 48.2 < 48.4

Comparing digits from left to right (tens, ones, tenths, hundredths) confirms the sequence.

Key Concept

Applying the order of operations (PEMDAS) to evaluate decimal expressions and comparing relative decimal values.

Question 389Question →

Planetary orbital mechanics dictate that a planet's mean orbital speed depends directly on its distance from the Sun. Based on their orbital distances and Keplerian motion, arrange the following planets in order of their average orbital velocity around the Sun, from fastest to slowest.

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Answer

Mercury, Mars, Jupiter, Neptune (from fastest to slowest orbital velocity)

The correct sequence from fastest to slowest average orbital velocity is Mercury, Mars, Jupiter, and Neptune. In heliocentric orbits, orbital speed is inversely related to the square root of the distance from the Sun. Mercury is closest to the Sun (~0.39 AU) and travels fastest at ~47.4 km/s. Mars is the next closest (~1.52 AU) at ~24.1 km/s. Jupiter is farther out (~5.20 AU) at ~13.1 km/s, and Neptune is the farthest (~30.07 AU), traveling slowest at ~5.4 km/s.

Step-by-Step Solution

1

Apply the physical relationship between semi-major axis (distance from the Sun) and planetary orbital speed.

Planets closer to the Sun require higher orbital velocities to maintain stable orbits against stronger gravitational attraction, meaning orbital speed decreases monotonically as solar distance increases.

According to Kepler's Third Law and Newton's law of universal gravitation, orbital speed

v \approx \sqrt{\frac{GM_{\odot}}{r}}

.

2

Determine the relative distances of the given planets from the Sun in astronomical units (AU).

Mercury (~0.39 AU) < Mars (~1.52 AU) < Jupiter (~5.20 AU) < Neptune (~30.07 AU).

Planets must be sequenced by their spatial position from the Sun outward to establish speed ranking.

3

Order the planets from closest to farthest to determine the sequence from fastest to slowest orbital speed.

Mercury (~47.4 km/s) → Mars (~24.1 km/s) → Jupiter (~13.1 km/s) → Neptune (~5.4 km/s).

Shortest distance corresponds to highest velocity, and greatest distance corresponds to lowest velocity.

Key Concept

Planetary orbital velocity decreases with increasing distance from the Sun due to gravitational mechanics.

Estimated Time:1m 30s

Question 390Question →

Following a massive glacial retreat, a newly exposed expanse of bare continental bedrock undergoes primary succession and soil formation. Arrange the following processes in the correct chronological order, from the initial exposure of the bare rock to the development of a mature soil profile.

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Answer

The correct chronological sequence is: mechanical fracturing, chemical breakdown by pioneer organisms, establishment of an organic surface horizon, downward mobilization of fine particles (eluviation), and accumulation of translocated minerals (illuviation).

Soil formation follows a logical progression driven by physical, biological, and chemical factors over time. It begins with mechanical weathering breaking down solid parent material. This allows pioneer organisms to colonize and initiate chemical weathering. As these organisms die, they add organic matter to create topsoil. Finally, water percolating through this topsoil leaches minerals (eluviation) and deposits them deeper down (illuviation) to form a mature subsoil horizon.

Step-by-Step Solution

1

Identify the starting condition.

The process begins with bare, unweathered bedrock.

Before biological or advanced chemical processes can occur, the solid rock must be physically broken down (mechanical weathering) to increase surface area.

2

Determine the initial biological interaction.

Pioneer organisms colonize the fragmented rock.

Lichens and mosses are the first to grow on bare rock, secreting acids that initiate chemical weathering.

3

Sequence the formation of the first soil layer.

Organic matter accumulates to form topsoil.

As pioneer organisms die and decompose, their organic matter mixes with the weathered mineral grains, creating an initial topsoil horizon.

4

Sequence the movement of materials within the profile.

Eluviation (leaching) occurs before illuviation (accumulation).

Water percolating through the new topsoil must first leach and dissolve minerals from the upper layers before those materials can be deposited deeper in the profile to form a subsoil horizon.

Key Concept

Pedogenesis and Soil Profile Development

Question 391Question →

As the Moon orbits Earth over roughly 29.5 days, observers on Earth see changing amounts of its sunlit side. Arrange the following four primary lunar phases in chronological order, starting from the beginning of a lunar cycle.

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Answer

The correct chronological sequence from the start of the lunar cycle is New Moon, followed by First Quarter, then Full Moon, and finally Third (Last) Quarter.

A complete lunar cycle begins with the New Moon, progresses through the waxing stages to the First Quarter, reaches the opposite side of Earth as a Full Moon, and then wanes through the Third Quarter before returning to a New Moon.

Step-by-Step Solution

1

Identify the phase that begins the lunar cycle.

The New Moon occurs at the start of the cycle (

0\text{ days}

) when the Moon aligns between Earth and the Sun.

The synodic month is measured starting from conjunction (New Moon).

2

Track the progression of illuminated phases as the Moon orbits counter-clockwise around Earth.

The Moon waxes through crescent phases to reach the First Quarter at approximately

7.4\text{ days}

, reaches Full Moon at approximately

14.8\text{ days}

, and wanes to the Third Quarter at approximately

22.1\text{ days}

.

The Moon orbits Earth in a counter-clockwise direction (viewed from above the North Pole), progressing through waxing phases before waning phases.

Key Concept

Lunar phase progression throughout a synodic month

Estimated Time:45s

Question 392Question →

In the human excretory system, liquid waste is filtered from the bloodstream and eliminated from the body. What is the correct sequence of organs through which urine travels from its production to excretion?

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Answer

The correct sequence from production to excretion is Kidneys, Ureters, Urinary bladder, and Urethra.

Urine formation begins in the functional units (nephrons) of the kidneys. Once formed, urine drains through the renal pelvis into the two ureters, which transport it downward to the urinary bladder for temporary storage. When the bladder contracts during urination, urine is discharged out of the body through the urethra.

Step-by-Step Solution

1

Identify the site of urine formation.

Kidneys filter blood to produce urine.

Urine production begins in the microscopic nephrons of the kidneys.

2

Trace the pathway from the kidneys toward storage.

Urine travels through the ureters.

The ureter tubes directly conduct urine from the renal pelvis down to the bladder.

3

Identify the temporary storage organ.

Urine enters the urinary bladder.

The bladder expands to hold urine until micturition occurs.

4

Determine the final duct for excretion.

Urine exits through the urethra.

The urethra serves as the single final passage connecting the bladder to the exterior.

Key Concept

Urinary system anatomical pathway and excretion mechanism

Estimated Time:45s

Question 393Question →

Beginning immediately after a New Moon, in what chronological order do the following lunar phases appear to an observer on Earth during a single monthly orbital cycle?

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Answer

Waxing Crescent → First Quarter → Full Moon → Third Quarter

Directly after a New Moon, the Moon moves along its orbit so that sunlight begins striking the side facing Earth, creating a Waxing Crescent. About one week into the roughly 29.5-day cycle, one-quarter of the orbit is complete, displaying the First Quarter (half lit). At approximately two weeks, the Moon reaches the opposite side of Earth from the Sun, producing a Full Moon. By the third week, as the illuminated fraction decreases, the Moon reaches the Third Quarter phase before finishing its cycle.

Step-by-Step Solution

1

Establish the starting point and direction of the lunar phase cycle

The cycle starts at New Moon (day 0), when the Moon lies between Earth and the Sun and is not illuminated from Earth's viewpoint.

Tracking phase order requires advancing sequentially from the designated starting condition.

2

Sequence the waxing phases leading up to maximum illumination

As the Moon travels counterclockwise in its orbit, visible illumination expands from a Waxing Crescent (days 1–6) to First Quarter (day 7, half illuminated), then continues expanding toward Full Moon (day 14, fully illuminated).

The term 'waxing' denotes growing illumination during the first half of the synodic month.

3

Sequence the waning phases following maximum illumination

After the Full Moon, the illuminated portion diminishes through Waning Gibbous to Third Quarter (day 21, half illuminated) and Waning Crescent before returning to New Moon.

The term 'waning' denotes shrinking illumination during the second half of the synodic month.

Key Concept

Lunar phase cycle and relative orbital geometry of the Earth, Moon, and Sun

Estimated Time:1m 0s

Question 394Question →

A logistics quartermaster is auditing the remaining capacity of four different fuel storage tanks relative to their maximum volume. To plan refueling priorities, arrange the tanks from the least full to the most full based on their current capacity levels.

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Answer

The correct order from least full to most full is Tank C (0.675), Tank D (17/25), Tank B (68.5%), and Tank A (11/16).

By converting all capacity values to percentages (

68.75\%

,

68.5\%

,

67.5\%

, and

68\%

), their magnitudes can be directly and accurately compared. The lowest percentage is

67.5\%

(Tank C), followed closely by

68\%

(Tank D),

68.5\%

(Tank B), and the highest is

68.75\%

(Tank A).

Step-by-Step Solution

1

Convert all given values to a common format (such as percentages or decimals) for easy and accurate comparison.

Tank C is given as a decimal,

0.675

, which equals

67.5\%

. Tank B is already given in percentage format at

68.5\%

.

Comparing mixed forms (fractions, decimals, and percentages) is prone to estimation errors without standardizing them.

2

Convert the fraction for Tank D into a percentage.

Tank D is

\frac{17}{25}

. Multiplying both the numerator and denominator by

4

creates a fraction out of

100

:

\frac{68}{100}

, which equals

0.68

or

68\%

.

Converting fractions to a base of

100

makes percentage translation immediate.

3

Convert the fraction for Tank A into a percentage using long division or known fractional equivalents.

Tank A is

\frac{11}{16}

. Since

\frac{1}{16} = 0.0625

, multiplying by

11

yields

0.6875

. Alternatively,

11 \div 16 = 0.6875

. This equates to

68.75\%

.

Finding the exact decimal prevents mistakes when comparing closely clustered values.

4

Order the standardized percentages sequentially from least to greatest.

67.5\% < 68\% < 68.5\% < 68.75\%

. Matching these values back to their original forms yields: Tank C, Tank D, Tank B, Tank A.

The prompt asks to arrange the tanks specifically from least full to most full.

Key Concept

Converting fractions and decimals to percentages to compare close magnitudes accurately.

Question 395Question →

Identify the correct step-by-step sequence to fully simplify the given mathematical expression. Order the operations from first to last:

15 - 3 \times (2 + 1)^2

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Answer

The sequence is to evaluate the parentheses first, apply the exponent to that result, perform the multiplication, and finally execute the subtraction.

Following the standard order of operations (PEMDAS), you must first resolve the grouping symbols (parentheses), then evaluate the exponent, perform the multiplication, and finally execute the subtraction.

Step-by-Step Solution

1

Evaluate the terms inside the parentheses:

(2 + 1)

3

Parentheses (grouping symbols) hold the highest priority in the order of operations.

2

Apply the exponent to the result:

3^2

9

Exponents must be evaluated before multiplication and subtraction.

3

Perform the multiplication:

3 \times 9

27

Multiplication is prioritized before subtraction.

4

Perform the subtraction:

15 - 27

-12

Subtraction is the final remaining operation in the expression.

Key Concept

Order of Operations

Estimated Time:45s

Question 396Question →

Arrange the following fractions and mixed numbers in ascending order (from least to greatest).

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Answer

The correct ascending order is

\frac{14}{9}

,

1\frac{5}{8}

,

1\frac{2}{3}

, and

\frac{7}{4}

.

By converting all values to improper fractions and finding a common denominator of 72, the values become

\frac{112}{72}

,

\frac{117}{72}

,

\frac{120}{72}

, and

\frac{126}{72}

. Comparing the numerators allows us to determine the correct ascending order of the original values.

Step-by-Step Solution

1

Convert all mixed numbers to improper fractions.

1\frac{5}{8} = \frac{13}{8}

and

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

.

Improper fractions are easier to evaluate when finding a common denominator.

2

Determine the least common multiple (LCM) of the denominators 9, 8, 3, and 4.

The LCM is 72.

A common denominator is required to accurately compare the sizes of fractions.

3

Convert each fraction into an equivalent fraction with a denominator of 72.

\frac{14}{9} = \frac{112}{72}

,

\frac{13}{8} = \frac{117}{72}

,

\frac{5}{3} = \frac{120}{72}

,

\frac{7}{4} = \frac{126}{72}

.

Having a common denominator allows for direct comparison of the numerators.

4

Compare the numerators to order the fractions from least to greatest.

112 < 117 < 120 < 126

, which means

\frac{112}{72} < \frac{117}{72} < \frac{120}{72} < \frac{126}{72}

.

The question specifies arranging the values in ascending order.

5

Map the ordered fractions back to their original forms.

\frac{14}{9}

,

1\frac{5}{8}

,

1\frac{2}{3}

,

\frac{7}{4}

.

To provide the final answer using the exact values given in the prompt.

Key Concept

Comparing and ordering mixed numbers and improper fractions with unlike denominators by finding a common denominator.

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