All practice questions

Question 361Question →

A naval technician inspects a solid aluminum component intended for an aircraft assembly. The component has a mass of $162\text{ g}$ and the density of aluminum is $2.7\text{ g/cm}^3$ . What is the volume of the component in cubic centimeters ( $\text{cm}^3$ )?

Show answer & explanation

Answer: 60

Answer

The volume of the aluminum component is

60\text{ cm}^3

.

Density is defined by the formula

\rho = \frac{m}{V}

. Rearranging algebraically to solve for volume yields

V = \frac{m}{\rho}

. Substituting the given values:

V = \frac{162\text{ g}}{2.7\text{ g/cm}^3} = 60\text{ cm}^3

.

Step-by-Step Solution

1

Identify the relevant physical formula and isolate the unknown variable

\text{Volume} = \frac{\text{Mass}}{\text{Density}}

Density (

\rho

) is a characteristic physical property defined as mass per unit volume (

\rho = \frac{m}{V}

).

2

Substitute the measured mass and known density to calculate volume

\text{Volume} = \frac{162\text{ g}}{2.7\text{ g/cm}^3} = 60\text{ cm}^3

Dividing

162

by

2.7

gives

60

, with units simplifying to

\text{cm}^3

.

Key Concept

Density as a fundamental physical property relating mass and volume

Estimated Time:1m 0s

Question 362Question →

A motorized winch pulls a $250\text{ kg}$ equipment sled across a level, frictionless sheet of ice. If the winch exerts a constant horizontal force of $750\text{ N}$ starting from rest, what is the speed of the sled, in meters per second ( $\text{m/s}$ ), after $4\text{ seconds}$ ?

Show answer & explanation

Answer: 12

Answer

The sled reaches a speed of

12\text{ m/s}

after

4\text{ seconds}

.

Applying Newton's second law gives an acceleration of

a = \frac{F}{m} = \frac{750\text{ N}}{250\text{ kg}} = 3\text{ m/s}^2

. Since the sled begins at rest (

v_0 = 0

), its speed after

4\text{ s}

is

v = at = 3\text{ m/s}^2 \times 4\text{ s} = 12\text{ m/s}

.

Step-by-Step Solution

1

Determine the acceleration produced by the net horizontal force

a = \frac{F_{\text{net}}}{m} = \frac{750\text{ N}}{250\text{ kg}} = 3\text{ m/s}^2

According to Newton's second law of motion (

F = ma

), acceleration equals the net force divided by mass.

2

Calculate final speed from initial rest and constant acceleration over time

v = v_0 + at = 0\text{ m/s} + (3\text{ m/s}^2)(4\text{ s}) = 12\text{ m/s}

Uniform acceleration kinematic equation relates velocity change to acceleration multiplied by elapsed time.

Key Concept

Combining Newton's second law (

F = ma

) with linear kinematics (

v = v_0 + at

) to determine final velocity under constant net force.

Alternative Method

Use the impulse-momentum theorem:

\text{Impulse} = F \cdot \Delta t = \Delta p = m \cdot v

. Substituting known values gives

(750\text{ N})(4\text{ s}) = (250\text{ kg})(v) \implies 3,000\text{ N}\cdot\text{s} = 250\text{ kg} \cdot v \implies v = \frac{3,000}{250} = 12\text{ m/s}

.

Estimated Time:1m 0s

Question 363Question →

A logistics convoy is assigned an $80$ -mile supply route. During the morning phase, the convoy travels $22 \frac{3}{8}$ miles. In the afternoon, they travel an additional $18 \frac{4}{5}$ miles before halting for a security check. After resuming, they cover another $14 \frac{1}{2}$ miles before nightfall. How many miles remain to complete the entire route? (Enter your answer as a decimal)

Show answer & explanation

Answer: 24.325

Answer

24.325

The total distance traveled is calculated by converting the fractional parts of the mixed numbers to a common denominator of 40. Adding

22 \frac{15}{40} + 18 \frac{32}{40} + 14 \frac{20}{40}

gives

55 \frac{27}{40}

miles. Subtracting this from the

80

-mile total route leaves

24 \frac{13}{40}

miles. Converting

\frac{13}{40}

to a decimal yields 0.325, giving a final answer of 24.325.

Step-by-Step Solution

1

Determine the least common denominator (LCD) for the fractional parts of the distances traveled.

The denominators are 8, 5, and 2. The LCD is 40.

Fractions must have a common denominator before they can be added together.

2

Convert each mixed number to have the common denominator of 40 and add them to find the total distance traveled.

22 \frac{15}{40} + 18 \frac{32}{40} + 14 \frac{20}{40} = 54 \frac{67}{40}

, which simplifies to

55 \frac{27}{40}

miles.

Adding the individual segments provides the total distance covered so far.

3

Subtract the total distance traveled from the overall route distance to find the remaining distance.

80 - 55 \frac{27}{40} = 79 \frac{40}{40} - 55 \frac{27}{40} = 24 \frac{13}{40}

miles.

Subtracting the part completed from the whole gives the remaining distance.

4

Convert the remaining distance into a decimal format.

\frac{13}{40} = 0.325

, making the final distance

24.325

miles.

The question specifically asks for the answer to be entered as a decimal.

Key Concept

Adding mixed numbers with unlike denominators and subtracting from a whole number.

Question 364Question →

A light training aircraft with a mass of $750\text{ kg}$ accelerates from rest along a straight runway under a constant net forward force of $1{}500\text{ N}$ . What is the velocity of the aircraft, in meters per second ( $\text{m/s}$ ), after $8\text{ seconds}$ ?

Show answer & explanation

Answer: 16

Answer

16 m/s

According to Newton's second law (

F = ma

), a net force of

1{}500\text{ N}

applied to a mass of

750\text{ kg}

produces a constant acceleration of

a = \frac{1{}500}{750} = 2\text{ m/s}^2

. Starting from rest (

v_0 = 0

), the velocity after

8\text{ seconds}

is

v = at = 2\text{ m/s}^2 \times 8\text{ s} = 16\text{ m/s}

.

Step-by-Step Solution

1

Calculate the constant acceleration of the aircraft using Newton's second law of motion (

F_{\text{net}} = ma

).

a = \frac{1{}500\text{ N}}{750\text{ kg}} = 2\text{ m/s}^2

Acceleration is directly proportional to net force and inversely proportional to mass.

2

Calculate the final velocity after

8\text{ seconds}

using the kinematic equation for uniform acceleration (

v = v_0 + at

).

v = 0\text{ m/s} + (2\text{ m/s}^2)(8\text{ s}) = 16\text{ m/s}

Starting from rest, the final velocity equals the constant acceleration multiplied by elapsed time.

Key Concept

Newton's Second Law and Uniform Acceleration

Alternative Method

Apply the impulse-momentum theorem:

\text{Impulse} = F_{\text{net}} \times \Delta t = \Delta p = m \times v

. Solving for velocity gives

v = \frac{F_{\text{net}} \times \Delta t}{m} = \frac{1{}500\text{ N} \times 8\text{ s}}{750\text{ kg}} = \frac{12{}000}{750} = 16\text{ m/s}

.

Estimated Time:1m 0s

Question 365Question →

A maintenance technician needs to cut a continuous length of electrical wire into smaller segments. If the technician has a piece of wire that is $45$ feet long and each segment must be $3 \frac{3}{4}$ feet long, how many full segments can be cut?

Show answer & explanation

Answer: 12

Answer

12

To find the total number of segments, the total length of wire (

45

feet) must be divided by the length of each individual segment (

3 \frac{3}{4}

feet). Converting

3 \frac{3}{4}

to the improper fraction

\frac{15}{4}

and multiplying

45

by its reciprocal (

\frac{4}{15}

) yields exactly

12

segments.

Step-by-Step Solution

1

Convert the mixed number to an improper fraction.

3 \frac{3}{4} = \frac{15}{4}

Division operations with mixed numbers require converting them to improper fractions first.

2

Set up the division equation.

45 \div \frac{15}{4}

To find how many smaller segments fit into the total length, you must divide the total wire length by the length of one individual segment.

3

Convert the division into multiplication.

45 \times \frac{4}{15}

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

4

Simplify the expression and calculate the final value.

3 \times 4 = 12

Cross-canceling

45

and

15

leaves a factor of

3

, which is then multiplied by

4

to reach the final whole number answer.

Key Concept

Dividing a whole quantity by a mixed number using improper fractions and reciprocals.

Question 366Question →

A maintenance crew can paint $120$ square feet of a ship's hull in $2$ hours. Working at this same constant rate, how many square feet can the crew paint in $5$ hours?

Show answer & explanation

Answer: 300

Answer

300

Because the work happens at a constant rate, the area painted is directly proportional to the time spent working. Finding the unit rate (

60

square feet per hour) and multiplying it by the target time (

5

hours) yields the correct total of

300

square feet.

Step-by-Step Solution

1

Calculate the unit rate of painting.

60

square feet per hour

To find how much is painted in a different amount of time, we first need to know how much is painted in one hour. We do this by dividing the total area (

120

) by the total time (

2

).

2

Calculate the total area painted in the new timeframe.

300

square feet

Multiply the unit rate (

60

square feet per hour) by the new target time (

5

hours) to find the final proportional amount.

Key Concept

Direct Proportions

Question 367Question →

An office manager is purchasing a commercial printer with a regular retail price of $\$1,200.00$ . The store is offering a $20\%$ discount on all electronics. Using a corporate account, the manager receives an additional $10\%$ discount applied to the reduced sale price. Finally, a $5\%$ sales tax is added to the discounted total. What is the final total cost of the printer in dollars?

Show answer & explanation

Answer: 907.2

Answer

The final total cost of the printer is

\$907.20

.

The correct answer is derived by first taking

20\%

off the

\$1,200.00

base price to get

\$960.00

. Then, the

10\%

corporate discount is taken off the

\$960.00

, bringing the subtotal down to

\$864.00

. Lastly, a

5\%

tax on the

\$864.00

subtotal adds

\$43.20

, resulting in a final cost of

\$907.20

.

Step-by-Step Solution

1

Calculate the price after the initial

20\%

electronics discount.

\$1,200.00 - (\$1,200.00 \times 0.20) = \$1,200.00 - \$240.00 = \$960.00

The first discount is applied to the original retail price of the printer.

2

Calculate the price after the additional

10\%

corporate discount.

\$960.00 - (\$960.00 \times 0.10) = \$960.00 - \$96.00 = \$864.00

Successive discounts are calculated based on the previously reduced price, not the original price.

3

Calculate the

5\%

sales tax on the final discounted price and add it to the total.

\$864.00 \times 0.05 = \$43.20

. Total =

\$864.00 + \$43.20 = \$907.20

Sales tax is always applied to the final discounted sale price before checkout.

Key Concept

Successive discounts must be applied sequentially to the new reduced price at each step, and sales tax is computed on the final discounted price.

Question 368Question →

An automated wire-cutting machine processes $350$ meters of copper wiring in $20$ minutes. Operating at this constant rate, how many meters of wiring will the machine process in $1.5$ hours?

Show answer & explanation

Answer: 1575

Answer

1575

Because the machine operates at a constant rate, the relationship between time and wire processed is directly proportional. However, the time units differ between the given rate (

20

minutes) and the target time (

1.5

hours). First, convert

1.5

hours to

90

minutes. Then, set up the direct proportion

\frac{350 \text{ meters}}{20 \text{ minutes}} = \frac{x \text{ meters}}{90 \text{ minutes}}

. Solving for

x

by multiplying

350

by

90

and dividing by

20

yields exactly

1575

meters.

Step-by-Step Solution

1

Harmonize the time units by converting hours to minutes.

90

minutes

The given rate is in meters per minute, so the target time must also be in minutes to establish a valid equivalent ratio.

2

Set up a direct proportion equation.

\frac{350}{20} = \frac{x}{90}

The length of wire processed increases at a constant, directly proportional rate to the time operated.

3

Solve the proportion for the unknown value

x

.

x = 1575

Cross-multiplying and simplifying isolates

x

, calculating the total meters processed in the longer timeframe.

Key Concept

Direct proportions with unit harmonization

Question 369Question →

A plumber is installing copper piping and starts with a full pipe that measures $14 \frac{1}{8}$ feet in length. For the installation, he cuts off one section measuring $3 \frac{1}{2}$ feet and another section measuring $5 \frac{3}{8}$ feet. Assuming no material is lost to the saw blade during the cuts, what is the length of the remaining pipe? Enter your answer as a decimal.

Show answer & explanation

Answer: 5.25

Answer

The length of the remaining pipe is 5.25 feet.

First, the total length of the cut sections is found by getting a common denominator (

3 \frac{4}{8} + 5 \frac{3}{8} = 8 \frac{7}{8}

feet). To find the remaining pipe, this total is subtracted from the original length. Since

\frac{7}{8}

is larger than

\frac{1}{8}

, regrouping is necessary:

14 \frac{1}{8}

becomes

13 \frac{9}{8}

. Subtracting yields

5 \frac{2}{8}

, which simplifies to

5 \frac{1}{4}

. The decimal equivalent of

5 \frac{1}{4}

is 5.25.

Step-by-Step Solution

1

Find a common denominator for the two sections that were cut.

3 \frac{1}{2} = 3 \frac{4}{8}

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

2

Add the lengths of the two cut sections together.

3 \frac{4}{8} + 5 \frac{3}{8} = 8 \frac{7}{8}

feet

This determines the total amount of pipe removed from the original length.

3

Set up the subtraction to find the remaining pipe.

14 \frac{1}{8} - 8 \frac{7}{8}

Subtracting the removed amount from the original total gives the remaining balance.

4

Regroup the initial total to allow for fraction subtraction.

14 \frac{1}{8} = 13 + 1 \frac{1}{8} = 13 \frac{9}{8}

You cannot subtract

\frac{7}{8}

from

\frac{1}{8}

without borrowing a whole number (1 =

\frac{8}{8}

).

5

Subtract the whole numbers and the fractions.

13 \frac{9}{8} - 8 \frac{7}{8} = 5 \frac{2}{8}

feet, which simplifies to

5 \frac{1}{4}

feet.

This isolates the precise length of the pipe that is left.

6

Convert the simplified fraction into a decimal format.

5 \frac{1}{4} = 5.25

The question specifically requests the final answer to be entered as a decimal.

Key Concept

Fraction Addition and Subtraction in Context

Question 370Question →

An industrial supply company purchases a heavy-duty welding machine directly from the manufacturer for $\$2,400.00$ . The company applies a $40\%$ markup to establish their standard retail price. During a promotional event, a customer purchases the machine at a $15\%$ discount off the retail price. If an additional $6\%$ sales tax is applied to the discounted price, what is the total amount the customer pays in dollars?

Show answer & explanation

Answer: 3027.36

Answer

3027.36

To find the final amount, each percentage change must be applied sequentially to the most recent price. First, the

40\%

markup increases the

\$2,400.00

base price to

\$3,360.00

. Next, the

15\%

discount reduces this retail price to

\$2,856.00

. Finally, applying the

6\%

sales tax to the discounted price yields a final total of

\$3,027.36

.

Step-by-Step Solution

1

Calculate the markup amount and the standard retail price.

\$2,400.00 \times 1.40 = \$3,360.00

The

40\%

markup is based on the initial wholesale cost.

2

Calculate the sale price after the discount.

\$3,360.00 \times (1 - 0.15) = \$3,360.00 \times 0.85 = \$2,856.00

The

15\%

discount is taken off the standard retail price, not the original cost.

3

Calculate the final price including sales tax.

\$2,856.00 \times 1.06 = \$3,027.36

Sales tax is calculated on the discounted sale price to find the total amount paid.

Key Concept

Calculating successive percentage changes (markups, discounts, and taxes) by updating the base value at each step.

Question 371Question →

A military water purification unit has three filters operating at different unit rates. Filter X can purify $180$ gallons of water in $1.5$ hours. Filter Y can purify $320$ gallons in $2$ hours. Filter Z's purification rate is exactly $20\%$ faster than Filter X's rate. If all three filters are turned on at the same time, how many minutes will it take them to completely purify $1,484$ gallons of water?

Show answer & explanation

Answer: 210

Answer

210

The correct answer is found by first determining the individual hourly rates of each filter, adding them together for a combined rate, calculating the total hours required for the given volume, and finally converting that time into minutes.

Step-by-Step Solution

1

Calculate the unit rate for Filter X.

120

gallons per hour

To find the combined rate later, we first need the individual hourly rate of Filter X.

180 \div 1.5 = 120

.

2

Calculate the unit rate for Filter Y.

160

gallons per hour

We need the individual hourly rate of Filter Y to find the total combined rate.

320 \div 2 = 160

.

3

Calculate the unit rate for Filter Z.

144

gallons per hour

Filter Z operates

20\%

faster than Filter X, so its rate is

120 + (0.20 \times 120) = 144

.

4

Calculate the combined unit rate of all three filters.

424

gallons per hour

Since the filters operate simultaneously, their individual rates add together:

120 + 160 + 144 = 424

.

5

Determine the time in hours needed to purify

1,484

gallons.

3.5

hours

Time equals the total amount required divided by the combined rate:

1,484 \div 424 = 3.5

.

6

Convert the total time from hours to minutes.

210

minutes

The question explicitly asks for the answer in minutes.

3.5 \times 60 = 210

.

Key Concept

Calculating, modifying, and combining multiple unit rates, followed by a final unit conversion.

Alternative Method

You could convert the rates to gallons per minute immediately. Filter X is 2 gallons per minute, Filter Y is 2.67 gallons per minute, etc. However, this introduces repeating decimals early on, making the math significantly harder without a calculator. Keeping the rates in hours avoids complex fractions until the final step.

Estimated Time:3m 30s

Question 372Question →

A machinist measures the mass of an irregular steel fastener to be $84\text{ g}$ . To determine its volume, the fastener is submerged in a graduated cylinder filled with $50\text{ mL}$ of water, causing the water level to rise to $62\text{ mL}$ . What is the density of the steel fastener in $\text{g/cm}^3$ ?

Show answer & explanation

Answer: 7

Answer

The density of the steel fastener is

7\text{ g/cm}^3

.

Density is calculated by dividing mass by volume (

\text{Density} = \frac{\text{Mass}}{\text{Volume}}

). The volume of the irregular fastener is found by subtracting the initial water volume from the final water volume (

62\text{ mL} - 50\text{ mL} = 12\text{ cm}^3

). Dividing the mass of

84\text{ g}

by the volume of

12\text{ cm}^3

gives

7\text{ g/cm}^3

.

Step-by-Step Solution

1

Determine the volume of the fastener using the water displacement method

Displaced volume =

62\text{ mL} - 50\text{ mL} = 12\text{ mL} = 12\text{ cm}^3

The volume of an irregular submerged solid equals the volume of fluid it displaces, where

1\text{ mL} = 1\text{ cm}^3

.

2

Calculate density by dividing mass by displaced volume

\text{Density} = \frac{84\text{ g}}{12\text{ cm}^3} = 7\text{ g/cm}^3

Density is defined as mass per unit volume (

\rho = \frac{m}{V}

).

Key Concept

Density and Water Displacement

Estimated Time:1m 0s

Question 373Question →

An industrial textile loom weaves $4.5$ yards of heavy canvas every $12$ minutes. Operating at this same constant rate, how many feet of canvas will the loom produce during a complete $8$ -hour work shift?

Show answer & explanation

Answer: 540

Answer

540 feet

To solve this problem, you must properly scale a direct proportion while paying careful attention to units. First, harmonize the time units: an 8-hour shift is equal to 480 minutes (8 × 60). Next, set up a proportion or find the scale factor: 480 minutes is exactly 40 times longer than the 12-minute baseline (480 ÷ 12 = 40). Because the loom operates at a constant rate, it will produce 40 times as much canvas. Multiply the 4.5 yards by 40 to find the total yards produced (4.5 × 40 = 180 yards). Finally, the question asks for the answer in feet. Since there are 3 feet in 1 yard, multiply 180 yards by 3 to get exactly 540 feet.

Step-by-Step Solution

1

Convert the total operating time from hours to minutes to match the given rate's unit of time.

8 \text{ hours} \times 60 \text{ minutes/hour} = 480 \text{ minutes}

Units of time must be identical before calculating the direct proportion.

2

Determine how many 12-minute intervals occur in the 480-minute shift.

480 \div 12 = 40 \text{ intervals}

This establishes the multiplier (scale factor) that will be applied to the fabric production.

3

Calculate the total yards of canvas produced during the entire shift.

40 \times 4.5 \text{ yards} = 180 \text{ yards}

Because production is directly proportional to time, multiplying the base production by the time scale factor yields total yards.

4

Convert the total production from yards to feet to answer the specific question asked.

180 \text{ yards} \times 3 \text{ feet/yard} = 540 \text{ feet}

The stem explicitly asks for the final length in feet, making this unit conversion a critical final step.

Key Concept

Solving multi-step direct proportion problems that require initial and final unit harmonizations (time and length).

Question 374Question →

Sarah's family goes out for dinner, and the total food bill is $60.00. They decide to leave a 15% tip for the server. How much money, in dollars, do they leave for the tip?

Show answer & explanation

Answer: 9

Answer

9

The tip amount is found by calculating 15% of

60.00. By multiplying

60.00

by

0.15

, the result is

9.00$, which represents the tip amount in dollars.

Step-by-Step Solution

1

Convert the tip percentage into a decimal.

15\% = 0.15

Percentages must be converted to decimals to perform multiplication.

2

Multiply the total bill by the decimal tip rate.

60.00 \times 0.15 = 9.00

The tip amount is a percentage of the total bill.

Key Concept

Calculating a percentage of a given monetary amount.

Question 375Question →

A carpenter is purchasing a new circular saw. The regular retail price of the saw is $\$240.00$ . If the hardware store is offering a $25\%$ discount during a holiday sale, what is the final sale price of the circular saw in dollars?

Show answer & explanation

Answer: 180

Answer

180

The final sale price is determined by subtracting the discount amount from the original price. The discount is calculated as

25\%

of

\$240.00

, which equals

\$60.00

. Subtracting

\$60.00

from

\$240.00

gives a final sale price of

\$180.00

.

Step-by-Step Solution

1

Convert the percentage to a decimal to calculate the discount amount.

25\% = 0.25

Percentages must be converted to decimals or fractions to perform arithmetic calculations.

2

Multiply the regular price by the decimal to find the total discount.

\$240.00 \times 0.25 = \$60.00

The discount is

25\%

of the original price.

3

Subtract the discount from the original price to find the final sale price.

\$240.00 - \$60.00 = \$180.00

The final sale price is the original price reduced by the discount amount.

Key Concept

Calculating percentage discounts and subtracting them from the original price.

Question 376Question →

A neutral isotope of zinc has a nucleus containing $36$ neutrons and has an overall mass number of $66$ . During a chemical reaction, this atom forms a $\text{Zn}^{2+}$ cation by losing $2$ electrons. Calculate the total number of electrons present in the $\text{Zn}^{2+}$ ion.

Show answer & explanation

Answer: 28

Answer

28 electrons

The mass number (

66

) minus the number of neutrons (

36

) gives an atomic number of

30

, which corresponds to

30

protons. A neutral atom has

30

electrons, and losing

2

electrons to become

\text{Zn}^{2+}

leaves exactly

28

electrons.

Step-by-Step Solution

1

Calculate the number of protons (atomic number) in the zinc atom

30 protons

The mass number equals the sum of protons and neutrons (

A = Z + N

), so

Z = 66 - 36 = 30

.

2

Calculate the remaining electrons in the

\text{Zn}^{2+}

cation

28 electrons

A neutral atom has an equal number of protons and electrons (30). A

+2

charge indicates the loss of 2 electrons (

30 - 2 = 28

).

Key Concept

Subatomic particle counts in isotopes and monatomic ions

Estimated Time:1m 0s

Question 377Question →

A naval shipyard receives a shipment of $945$ tons of specialized steel plating. The shipyard superintendent allocates $\frac{4}{9}$ of this shipment specifically for submarine hull repairs. If each submarine repair module requires exactly $17 \frac{1}{2}$ tons of steel plating, how many complete repair modules can be outfitted from the allocated steel?

Show answer & explanation

Answer: 24

Answer

24

The correct answer is found by first calculating the portion of the steel available for repairs (

945 \times \frac{4}{9} = 420

tons), and then dividing that available amount by the requirement per module (

420 \div 17 \frac{1}{2} = 420 \times \frac{2}{35} = 24

complete modules).

Step-by-Step Solution

1

Determine the total amount of steel allocated for submarine repairs.

420 tons

The problem states that only

\frac{4}{9}

of the

945

ton shipment is allocated for this specific purpose, requiring multiplication:

945 \times \frac{4}{9} = 105 \times 4 = 420

.

2

Convert the mixed number representing the steel required per module into an improper fraction.

\frac{35}{2}

tons

Converting

17 \frac{1}{2}

to an improper fraction (

\frac{17 \times 2 + 1}{2} = \frac{35}{2}

) is necessary to accurately divide the total allocated steel.

3

Calculate the number of complete modules by dividing the allocated steel by the required steel per module.

24 modules

Dividing the total allocated amount (

420

tons) by the amount needed per module (

\frac{35}{2}

tons) gives the number of modules:

420 \div \frac{35}{2} = 420 \times \frac{2}{35} = 12 \times 2 = 24

.

Key Concept

Applying fraction multiplication to find a portion of a whole, followed by fraction division to determine how many equal groups can be formed.

Question 378Question →

A naval ship uses $400$ pounds of provisions to feed a specific crew for $5$ days. At this same constant rate, how many pounds of provisions are needed to feed the crew for $12$ days?

Show answer & explanation

Answer: 960

Answer

960

By determining the unit rate of

80

pounds per day and multiplying it by

12

days, the direct proportionality rule is correctly applied to find the total of

960

pounds.

Step-by-Step Solution

1

Calculate the unit rate of provisions per day.

400 \div 5 = 80

pounds per day.

Finding the amount used in one day allows scaling the proportion to any other number of days.

2

Multiply the unit rate by the new number of days.

80 \times 12 = 960

pounds.

This scales the daily rate up to the required

12

-day period to maintain the direct proportion.

Key Concept

Direct proportions using unit rates

Estimated Time:45s

Question 379Question →

An engine transmission system uses three interconnected gears: Gear A, Gear B, and Gear C. The ratio of the number of teeth on Gear A to the number of teeth on Gear B is $4:3$ . The ratio of the number of teeth on Gear B to the number of teeth on Gear C is $5:2$ . If Gear C has exactly $18$ teeth, what is the total number of teeth on all three gears combined?

Show answer & explanation

Answer: 123

Answer

123

The correct answer is found by linking the two separate proportions using their common element, Gear B. By converting the A:B ratio (

4:3

) and the B:C ratio (

5:2

) to share a common value for B (

15

), we get a unified, continuous ratio for A:B:C of

20:15:6

. The problem states that Gear C has

18

teeth. Since Gear C represents

6

parts in our combined ratio, each part must equal

3

teeth (

18 \div 6 = 3

). The total number of parts across all three gears is

41

(

20 + 15 + 6

). Multiplying the

41

total parts by

3

teeth per part results in a total of

123

teeth for the entire system.

Step-by-Step Solution

1

Align the separate ratios based on the common element.

Gear A : Gear B =

4:3

and Gear B : Gear C =

5:2

.

To compare all three gears simultaneously, Gear B must have the same value in both ratios.

2

Find a common multiple for the ratio numbers representing Gear B (

3

and

5

).

The least common multiple of

3

and

5

is

15

.

Creating a common denominator allows us to link the two independent proportions.

3

Scale both ratios so Gear B is represented by

15

parts.

Ratio A:B becomes

(4 \times 5):(3 \times 5) = 20:15

. Ratio B:C becomes

(5 \times 3):(2 \times 3) = 15:6

.

Multiplying both sides of a ratio by the same number maintains the proportion.

4

Combine into a single three-part ratio and determine total parts.

Combined ratio A:B:C is

20:15:6

. Total ratio parts =

20 + 15 + 6 = 41

.

This single ratio precisely maps the relationships between all three gears in the system.

5

Calculate the multiplier using Gear C's known teeth count.

6

parts =

18

teeth. Therefore,

1

part =

18 / 6 = 3

teeth.

This establishes the real-world value of a single ratio unit in this specific transmission system.

6

Calculate the total number of teeth on all gears combined.

41

total parts

\times 3

teeth per part =

123

teeth.

Multiplying the total number of parts by the value of a single part gives the final aggregate quantity.

Key Concept

Combining overlapping ratios by calculating the least common multiple of the shared term, then scaling the combined ratio to determine unknown totals.

Question 380Question →

A maintenance worker is painting transport vehicles. It takes $\frac{3}{5}$ of a gallon of paint to cover one vehicle. If the worker needs to paint $15$ identical vehicles, what is the total number of gallons of paint required?

Show answer & explanation

Answer: 9

Answer

9

The correct answer is found by taking the required amount of paint per vehicle (

\frac{3}{5}

) and multiplying it by the total number of vehicles (

15

). The calculation is

15 \times \frac{3}{5} = \frac{15 \times 3}{5} = \frac{45}{5} = 9

gallons.

Step-by-Step Solution

1

Set up the mathematical expression

15 \times \frac{3}{5}

To find the total amount of paint, multiply the amount of paint needed for a single vehicle by the total number of vehicles.

2

Calculate the final value

9

Multiply the whole number

15

by the numerator

3

to get

45

, then divide by the denominator

5

to get

9

.

Key Concept

Multiplying a whole number by a fraction to find a total quantity.

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