All practice questions

Determine the final sale price based on the given information.

Fill in the blanks below

A homeowner wants to install a new water heater. The regular price of the water heater is

\$600

. The hardware store offers a

20\%

discount during a weekend sale.

The final sale price of the water heater is $\$$.

Show answer & explanation

Answer

480

The regular price is

600 and the discount is 20%. To find the discount amount, multiply

600 by 0.20, which yields

120. Subtracting the

120 discount from the regular

600 price results in a final sale price of

480.

Step-by-Step Solution

1

Calculate the amount of the discount.

20% of

600 = 0.20 \times 600 =

120

The discount is a percentage of the regular price.

2

Subtract the discount amount from the regular price to find the final sale price.

600 -

120 = $480

A discount reduces the original cost of the item.

Key Concept

Calculating the final price after a simple percentage discount is applied.

Alternative Method

Instead of calculating the discount and subtracting it, you can multiply the regular price by the percentage of the price you are actually paying. If the discount is 20%, you pay 100% - 20% = 80% of the original price.

600 \times 0.80 =

480.

Estimated Time:45s

Question 3282Question →

A carpentry team is preparing materials for a project. They have a wooden beam that is $12$ feet long. They need to cut the beam into smaller support blocks that are each $\frac{3}{4}$ of a foot long. How many support blocks can they cut from the beam?

4

9

16

3

Show answer & explanation

Answer: $16$

Answer

The team can cut

16

support blocks from the beam.

The correct answer is

16

. To find how many smaller pieces can be cut from a larger piece, you must divide the total length by the length of one piece. Dividing

12

by

\frac{3}{4}

is the same as multiplying

12

by

\frac{4}{3}

, which results in

16

.

Step-by-Step Solution

1

Identify the total length of the beam and the length of each required block.

The total length is

12

feet, and each block is

\frac{3}{4}

of a foot long.

These values are needed to determine how many times the smaller length fits into the total length.

2

Set up a division equation.

12 \div \frac{3}{4}

To find how many equal pieces can be made from a total amount, division is required.

3

Convert the division problem into a multiplication problem by using the reciprocal of the fraction.

12 \times \frac{4}{3}

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

4

Multiply the whole number by the numerator and divide by the denominator.

\frac{12 \times 4}{3} = \frac{48}{3} = 16

This completes the calculation, showing the total number of blocks.

Key Concept

Fraction Division in Context

Question 3283Question →

An earth-moving excavator clears $120$ cubic yards of debris in $45$ minutes. Operating at this constant rate, how many cubic yards of debris will the excavator clear in $3$ hours?

480

8

30

48

Show answer & explanation

Answer: 480

Answer

480

The correct answer is found by first converting

3

hours into

180

minutes. Then, applying a direct proportion,

\frac{120}{45} = \frac{x}{180}

, yields

x = 480

. Alternatively, observing that

180

minutes is exactly

4

times as long as

45

minutes means the excavator will clear

4

times the debris:

120 \times 4 = 480

.

Step-by-Step Solution

1

Convert the target time from hours to minutes so the units match the given rate.

3 \text{ hours} \times 60 \text{ minutes/hour} = 180 \text{ minutes}

Units of time must be identical before setting up a valid proportion.

2

Set up a direct proportion relating the volume of debris to the time taken.

\frac{120}{45} = \frac{x}{180}

Since the excavator works at a constant rate, the ratio of volume to time remains constant.

3

Solve for the unknown volume (

x

) by cross-multiplying or finding a common multiplier.

x = \frac{120 \times 180}{45} = 120 \times 4 = 480

Isolating

x

provides the total cubic yards cleared in

180

minutes.

Key Concept

Direct proportions with unit conversions

Estimated Time:1m 0s

Question 3284Question →

A heavy-duty diesel generator has a maximum fuel capacity of $120$ gallons. The tank currently contains $84 \frac{1}{3}$ gallons of diesel. If a technician adds $18 \frac{3}{5}$ gallons of fuel to the tank, how many more gallons are needed to completely fill the tank?

(Enter your answer as a mixed number in its simplest form, using a space between the whole number and the fraction, such as 5 1/2)

Fill in the blanks below

The generator needs more gallons of fuel.

Show answer & explanation

Answer

17 1/15

By finding a common denominator of 15, the total fuel currently in the tank is calculated as

102 \frac{14}{15}

gallons. Subtracting this from the

120

-gallon total capacity requires borrowing

1

from

120

to make it

119 \frac{15}{15}

. Subtracting

102 \frac{14}{15}

from

119 \frac{15}{15}

leaves

17 \frac{1}{15}

gallons.

Step-by-Step Solution

1

Find a common denominator to add the current fuel and the added fuel together.

The lowest common multiple of 3 and 5 is 15.

84 \frac{1}{3} = 84 \frac{5}{15}

18 \frac{3}{5} = 18 \frac{9}{15}

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

2

Add the two mixed numbers to find the total amount of fuel currently in the tank.

84 \frac{5}{15} + 18 \frac{9}{15} = 102 \frac{14}{15}

gallons.

This establishes the current volume of fuel occupying the tank.

3

Subtract the total current fuel from the maximum capacity of the tank.

120 - 102 \frac{14}{15}

To subtract, borrow 1 from 120 to create a fraction:

119 \frac{15}{15}

.

119 \frac{15}{15} - 102 \frac{14}{15} = 17 \frac{1}{15}

gallons.

Subtracting the volume currently inside the tank from the maximum capacity gives the remaining space available.

Key Concept

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

Question 3285Question →

A specialized automotive repair shop orders a set of custom performance tires. The shop applies a $50\%$ markup to the wholesale cost to establish the regular customer price. During a seasonal promotion, the shop offers a $30\%$ discount off the regular customer price. If a customer pays $\$420.00$ for the set of tires during this promotion, what was the shop's original wholesale cost for the tires?

\$300.00

\$350.00

\$400.00

\$1,200.00

Show answer & explanation

Answer: \$400.00

Answer

The original wholesale cost for the tires was

\$400.00

.

To find the original wholesale cost, we must reverse the successive percentage operations. A

50\%

markup means the regular price is

150\%

of the wholesale cost (a

1.50

multiplier). A

30\%

discount means the customer pays

70\%

of the regular price (a

0.70

multiplier). Multiplying these factors gives a net multiplier of

1.50 \times 0.70 = 1.05

. This means the final price of

\$420.00

is exactly

105\%

of the wholesale cost. Dividing

\$420.00

by

1.05

yields

\$400.00

.

Step-by-Step Solution

1

Define the variable and set up the multipliers for successive percentage changes.

Let

W

be the wholesale cost. The markup multiplier is

(1 + 0.50) = 1.50

. The discount multiplier is

(1 - 0.30) = 0.70

.

A

50\%

markup increases the cost to

150\%

, and a

30\%

discount reduces the marked-up price to

70\%

of its value.

2

Create an equation relating the wholesale cost to the final customer price.

W \times 1.50 \times 0.70 = 420

Applying the successive multipliers to the wholesale cost must equal the final price paid by the customer.

3

Simplify the equation and solve for the wholesale cost.

1.05W = 420 \implies W = \frac{420}{1.05} = 400

Multiplying

1.50 \times 0.70

reveals that the final price is exactly

105\%

of the wholesale cost. Dividing isolating

W

provides the initial value.

Key Concept

Solving multi-step percentage problems by tracking successive multipliers backwards from a final price.

Question 3286Question →

Determine the correct numerical value to complete the statement below based on the provided rates.

Fill in the blanks below

An airborne logistics division drops

3.5

tons of humanitarian rations every

12

minutes. Operating at this constant rate, the division will drop pounds of rations in

1.5

hours.

*(Note:

1 \text{ ton} = 2,000 \text{ pounds}

)*

Show answer & explanation

Answer

52,500

The scenario requires calculating a proportional amount while successfully navigating two unit conversions. First, harmonize the time units by converting

1.5

hours to

90

minutes. Next, convert the initial weight of

3.5

tons into

7,000

pounds. Using these standardized units, set up the proportion:

7,000 \text{ lbs} / 12 \text{ min} = x \text{ lbs} / 90 \text{ min}

. Cross-multiplying gives

12x = 630,000

, and dividing by

12

results in exactly

52,500

pounds.

Step-by-Step Solution

1

Convert the target time frame from hours to minutes so it matches the time unit in the given rate.

1.5 \text{ hours} \times 60 \text{ minutes/hour} = 90 \text{ minutes}

Direct proportions require time units to be identical across both ratios.

2

Convert the initial weight from tons to pounds to match the requested unit for the final answer.

3.5 \text{ tons} \times 2,000 \text{ pounds/ton} = 7,000 \text{ pounds}

Harmonizing the weight units prevents conversion errors at the end of the calculation.

3

Set up a direct proportion equation comparing weight to time.

\frac{7,000 \text{ pounds}}{12 \text{ minutes}} = \frac{x \text{ pounds}}{90 \text{ minutes}}

Because the drop rate is constant, the ratio of weight to time remains equal.

4

Solve for the unknown weight (x).

x = \frac{7,000 \times 90}{12} = \frac{630,000}{12} = 52,500 \text{ pounds}

Cross-multiplying and dividing isolates the variable, yielding the total pounds dropped over the full time period.

Key Concept

Setting up and solving a direct proportion after performing multiple unit conversions.

Question 3287Question →

A customer is purchasing a pair of wireless headphones with a regular retail price of $\$150.00$ . If the store is running a weekend promotion offering a $20\%$ discount on all audio equipment, what is the final sale price of the headphones?

\$120.00

\$130.00

\$30.00

\$147.00

Show answer & explanation

Answer: $\$120.00$

Answer

\$120.00

The final sale price is determined by first finding

20\%

of the

\$150.00

regular price, which equals

\$30.00

. Subtracting that

\$30.00

discount from the regular price gives the final price of

\$120.00

.

Step-by-Step Solution

1

Calculate the dollar amount of the discount.

20\% \times \$150.00 = 0.20 \times 150 = \$30.00

To find the amount saved, convert the discount percentage to a decimal and multiply it by the regular retail price.

2

Subtract the discount amount from the regular retail price.

\$150.00 - \$30.00 = \$120.00

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

Key Concept

Calculating a single percentage discount and finding the new reduced total.

Question 3288Question →

In an occupational safety training session, workers are informed that exposure to steam at $100^\circ\text{C}$ causes significantly more severe burns than exposure to liquid water at $100^\circ\text{C}$ . Which thermodynamic factor directly accounts for this difference?

A higher average molecular kinetic energy in steam molecules than in liquid water molecules

The substantial release of latent heat as steam condenses into liquid upon contact with the skin

A faster transfer of thermal energy driven exclusively by electromagnetic radiation

A rapid exothermic chemical reaction occurring between water vapor and skin tissue

Show answer & explanation

Answer: The substantial release of latent heat as steam condenses into liquid upon contact with the skin

Answer

The substantial release of latent heat as steam condenses into liquid upon contact with the skin

When steam at

100^\circ\text{C}

comes in contact with cooler skin, it undergoes a phase change into liquid water at

100^\circ\text{C}

. During this condensation process, it releases its latent heat of vaporization (approximately

2{}260\text{ J/g}

) directly onto the skin before the resulting water even begins to cool. By contrast, boiling liquid water at

100^\circ\text{C}

only releases sensible heat as its temperature drops (

q = mc\Delta T

), yielding roughly

264\text{ J/g}

when cooling to body temperature. The additional latent heat released by steam transfers nearly ten times as much total energy to the skin, causing far more destructive burns.

Step-by-Step Solution

1

Analyze the thermal energy transferred by liquid water at

100^\circ\text{C}

Liquid water at

100^\circ\text{C}

transfers sensible heat only as it cools down to body temperature (

37^\circ\text{C}

), which equals approximately

264\text{ J/g}

.

Liquid water does not undergo a phase change when contacting skin.

2

Analyze the thermal energy transferred by steam at

100^\circ\text{C}

Steam at

100^\circ\text{C}

must first condense into liquid water at

100^\circ\text{C}

, releasing approximately

2{}260\text{ J/g}

of latent heat of condensation, and then release another

264\text{ J/g}

of sensible heat as the liquid cools.

Phase change from vapor to liquid requires the release of latent heat of vaporization.

3

Compare total energy delivered to the tissue

Steam delivers nearly ten times more thermal energy per gram than boiling water at the same temperature.

The large latent heat of condensation accounts directly for the greater severity of the burn.

Key Concept

Latent heat of phase change and heat transfer during condensation

Estimated Time:1m 0s

Question 3289Question →

Marcus purchases a tool set with a price tag of $40.00. If the hardware store applies a 5% sales tax, what is the final amount Marcus must pay?

$42.00

$45.00

$60.00

$2.00

Show answer & explanation

Answer: $42.00

Answer

The total cost of the tool set including tax is $42.00.

To find the final amount, you must calculate the sales tax and add it to the original price. 5% of

40.00 is

2.00. Adding this

2.00 tax to the

40.00 price tag yields a total of $42.00.

Step-by-Step Solution

1

Calculate the amount of sales tax by finding 5% of $40.00.

40.00 \times 0.05 =

2.00

Sales tax is a percentage of the original price.

2

Add the calculated sales tax to the original price to find the final total.

40.00 +

2.00 = $42.00

The total amount paid includes both the item's price and the tax applied to it.

Key Concept

Calculating sales tax as a percentage and adding it to the base price to determine the total cost.

Estimated Time:45s

Question 3290Question →

Read the scenario and fill in the blank with the correct numerical value.

Fill in the blanks below

A logistics team can pack

30

ration boxes in

15

minutes. Operating at this same constant rate, the team can pack ration boxes in

45

minutes.

Show answer & explanation

Answer

90

Because the packing rate is constant, the relationship between boxes and minutes is directly proportional. Determining the unit rate (

2

boxes per minute) and multiplying it by the target time (

45

minutes) correctly yields

90

boxes.

Step-by-Step Solution

1

Calculate the unit rate of packing.

30 \div 15 = 2

boxes per minute.

To find out how many boxes can be packed in a different amount of time, it helps to first know how many boxes are packed in exactly one minute.

2

Multiply the unit rate by the new time.

2 \times 45 = 90

boxes.

Since the team packs 2 boxes every minute, multiplying this constant rate by 45 minutes gives the total number of boxes packed in that timeframe.

Key Concept

Direct Proportions

Alternative Method

You can also solve this by setting up a direct proportion equation:

\frac{30}{15} = \frac{x}{45}

. Recognizing that

45

is exactly

3

times

15

, you can simply multiply the initial number of boxes by

3

:

30 \times 3 = 90

.

Estimated Time:45s

Question 3291Question →

A lunar rover has a mass of $90\text{ kg}$ on Earth. While operating on the lunar surface, where gravitational acceleration is approximately $1.6\text{ m/s}^2$ , its electric drive system generates a constant net forward force of $270\text{ N}$ . What is the forward acceleration of the rover?

18.0\text{ m/s}^2

3.0\text{ m/s}^2

4.6\text{ m/s}^2

0.3\text{ m/s}^2

Show answer & explanation

Answer: $3.0\text{ m/s}^2$

Answer

The forward acceleration of the rover is

3.0\text{ m/s}^2

.

According to Newton's second law of motion, acceleration is directly proportional to the net force applied and inversely proportional to the mass (

a = F_{\text{net}} / m

). Because mass is constant regardless of location, the rover's mass on the Moon is still

90\text{ kg}

. Dividing the net force of

270\text{ N}

by

90\text{ kg}

yields an acceleration of

3.0\text{ m/s}^2

.

Step-by-Step Solution

1

Identify the invariant mass of the rover

m = 90\text{ kg}

Mass measures the amount of matter in an object and remains constant regardless of local gravity.

2

Apply Newton's second law of motion (

F_{\text{net}} = ma

)

a = \frac{F_{\text{net}}}{m} = \frac{270\text{ N}}{90\text{ kg}} = 3.0\text{ m/s}^2

Horizontal acceleration depends solely on the net horizontal force divided by the object's mass.

Key Concept

Newton's Second Law of Motion and Mass Invariance

Estimated Time:1m 0s

Question 3292Question →

During a mass spectrometry investigation, researchers identify an isotope of chromium with a mass number of $52$ and an atomic number of $24$ . How many neutrons are contained in the nucleus of this chromium atom?

24

28

52

76

Show answer & explanation

Answer: $28$

Answer

28

neutrons

The mass number (

A = 52

) is the total number of protons and neutrons in the nucleus. The atomic number (

Z = 24

) indicates the number of protons. Subtracting the atomic number from the mass number (

52 - 24 = 28

) gives the total number of neutrons.

Step-by-Step Solution

1

Identify the given atomic parameters

Mass number (

A

) =

52

; Atomic number (

Z

) =

24

The mass number represents the sum of protons and neutrons, while the atomic number represents the number of protons.

2

Apply the neutron calculation formula:

\text{Neutrons} = A - Z

\text{Neutrons} = 52 - 24 = 28

Subtracting the number of protons from the total nucleon count isolates the number of neutrons.

Key Concept

Subatomic Particle Relationships in Atomic Structure

Estimated Time:45s

Question 3293Question →

A bakery owner is buying a new pastry display case. The regular price of the case is $\$800.00$ , but the supplier offers a $15\%$ discount for paying upfront. What is the final price of the display case?

\$120.00

\$785.00

\$680.00

\$788.00

Show answer & explanation

Answer: $\$680.00$

Answer

The final price of the display case is

\$680.00

.

The final price is found by first calculating the

15\%

discount amount (

\$120.00

) and then subtracting it from the original price (

\$800.00

), which results in

\$680.00

.

Step-by-Step Solution

1

Calculate the discount amount by finding

15\%

of the regular price.

\$800.00 \times 0.15 = \$120.00

To determine how much money will be taken off the original cost.

2

Subtract the discount amount from the regular price.

\$800.00 - \$120.00 = \$680.00

To find the final discounted sale price.

Key Concept

Calculating a percentage of a quantity and subtracting it to find a discounted price.

Alternative Method

Instead of calculating the discount and subtracting, you can multiply the regular price by the percentage you actually pay. Since the discount is

15\%

, the bakery owner pays

100\% - 15\% = 85\%

of the original price.

\$800.00 \times 0.85 = \$680.00

.

Estimated Time:45s

Question 3294Question →

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 3295Question →

Read the scenario and determine the final quantity. Fill in the blank with the correct numeric value.

Fill in the blanks below

A military engineering unit has a stockpile of

315

tons of crushed rock. They use

\frac{4}{9}

of the total stockpile to reinforce a main supply route. The unit commander then orders the remaining crushed rock to be divided into equal truckloads of

12 \frac{1}{2}

tons each for distribution to forward operating bases. The unit will be able to distribute exactly full truckloads to the forward bases.

Show answer & explanation

Answer

14

By first finding the remaining amount of crushed rock (

175

tons) and then dividing it by the size of each load (

12 \frac{1}{2}

tons), we determine there are exactly

14

truckloads.

Step-by-Step Solution

1

Determine the fraction of the stockpile that remains after the main supply route is reinforced.

1 - \frac{4}{9} = \frac{5}{9}

The truckloads are created from the remaining crushed rock, not the amount used for the main supply route.

2

Calculate the total weight of the remaining crushed rock.

315 \times \frac{5}{9} = \frac{315 \times 5}{9} = 35 \times 5 = 175

tons

This gives the exact quantity of rock available to be divided into individual truckloads.

3

Determine the number of truckloads by dividing the remaining weight by the weight per truckload.

175 \div 12 \frac{1}{2} = 175 \div \frac{25}{2} = 175 \times \frac{2}{25} = 7 \times 2 = 14

truckloads

Dividing the total available weight by the capacity of each load yields the total number of loads.

Key Concept

Solving multi-step word problems involving multiplication of whole numbers by fractions and division by mixed numbers.

Question 3296Question →

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 3297Question →

A customer purchases a tablet with an original price of $300.00. The tablet is on sale for 20% off. After the discount is applied, a 5% sales tax is added to the sale price. What is the final total cost of the tablet?

$241.20

$252.00

$255.00

$294.00

Show answer & explanation

Answer: $252.00

Answer

$252.00

To find the total cost, first find the sale price by calculating 20% of

300.00, which is

60.00. Subtract this discount from the original price to get

240.00. Next, calculate the 5% sales tax on this new sale price: 0.05 ×

240.00 =

12.00. Finally, add the tax to the sale price:

240.00 +

12.00 =

252.00.

Step-by-Step Solution

1

Calculate the discount amount.

20% of

300.00 = 0.20 ×

300.00 = $60.00

The discount must be determined to find the sale price.

2

Subtract the discount from the original price.

300.00 -

60.00 = $240.00

This provides the discounted sale price upon which tax is calculated.

3

Calculate the sales tax on the sale price.

5% of

240.00 = 0.05 ×

240.00 = $12.00

Sales tax is always applied to the final discounted price of the item.

4

Add the sales tax to the sale price.

240.00 +

12.00 = $252.00

This determines the final total cost the customer must pay.

Key Concept

Calculating sequential percentages in multi-step financial problems (applying discounts before taxes).

Alternative Method

You can solve this using multipliers in a single step: Total Cost = Original Price × (1 - Discount Rate) × (1 + Tax Rate). Here,

300.00 × 0.80 × 1.05 =

252.00.

Estimated Time:1m 30s

Question 3298Question →

Read the financial scenario below and fill in the missing monetary amounts.

Fill in the blanks below

A small business owner takes out a short-term loan to purchase new equipment. The loan accrues annual simple interest at a rate of

6\%

. After

9

months, the owner repays the loan in full with a single payment of

\$4,180

, which includes both the original principal and the accumulated interest.

The original principal amount of the loan was $\$

and the total interest charged was

\$$.

Show answer & explanation

Answer

The original principal amount was

4,000 and the interest charged was

180.

To find the original principal and interest from a total payoff amount, we must relate them using the formula

A = P(1 + RT)

. Because the rate is annual, the

9

months must be expressed as

0.75

years (or

3/4

). Substituting the knowns gives

4180 = P(1 + 0.06 \times 0.75)

, which simplifies to

4180 = 1.045P

. Dividing

4180

by

1.045

gives a principal of

\$4,000

. The interest is the total paid minus the principal:

4180 - 4000 = \$180

.

Step-by-Step Solution

1

Convert the time period from months to years.

T = 9 \text{ months} = \frac{9}{12} \text{ years} = \frac{3}{4} \text{ year}

Annual simple interest rates require the time variable to be in years.

2

Set up the simple interest total amount formula.

A = P(1 + RT)

This formula relates the known total payoff amount (

A

) directly to the unknown principal (

P

).

3

Substitute the known values and simplify the interest multiplier.

4180 = P(1 + 0.06 \times \frac{3}{4}) \Rightarrow 4180 = P(1 + 0.045) = 1.045P

To create a single linear equation with

P

as the only unknown.

4

Solve for the principal (

P

).

P = \frac{4180}{1.045} = 4000

Dividing the total amount by the interest factor isolates the initial loan amount.

5

Calculate the total interest charged (

I

).

I = A - P = 4180 - 4000 = 180

The interest is the difference between the total amount paid and the original principal.

Key Concept

Calculating principal from a total maturity value (Principal + Interest) using fractional years in a simple interest context.

Estimated Time:1m 30s

Question 3299Question →

A portable generator consumes $8$ liters of fuel every $12$ hours of operation. At this constant rate, how many liters of fuel will the generator consume in $48$ hours of operation?

2

3.2

32

72

Show answer & explanation

Answer: 32

Answer

The generator will consume 32 liters of fuel.

Because the generator operates at a constant rate, the relationship between time and fuel is a direct proportion. The new operating time (

48

hours) is exactly

4

times longer than the original operating time (

12

hours). Therefore, the fuel consumption will also be

4

times greater:

8 \text{ liters} \times 4 = 32 \text{ liters}

.

Step-by-Step Solution

1

Determine the relationship between the two operating times.

48 \div 12 = 4

Finding the scale factor makes it simple to calculate the proportionally larger amount of fuel.

2

Multiply the original fuel amount by the scale factor.

8 \times 4 = 32

liters

In a direct proportion, whatever factor scales the time must also scale the fuel consumption.

Key Concept

Direct proportions state that as one quantity increases by a certain multiplier, the related quantity increases by that exact same multiplier.

Question 3300Question →

In modern energy-efficient construction, a radiant barrier made of highly reflective aluminum foil is often installed in attics facing an open air gap. Which heat transfer mechanism is this reflective surface primarily designed to reduce?

Conduction through solid structural framing

Radiation emitted across the open air space

Convection generated by circulating attic air currents

Phase changes caused by moisture condensation

Show answer & explanation

Answer: Radiation emitted across the open air space

Answer

Radiation emitted across the open air space

Thermal radiation travels in the form of electromagnetic infrared waves across open spaces. Radiant barriers possess high thermal reflectivity and low emissivity, meaning they reflect the majority of radiant heat emitted by a hot roof deck rather than absorbing and reradiating it into the attic.

Step-by-Step Solution

1

Identify the physical properties and placement of the material described in the stem.

The material is a shiny, highly reflective foil facing an open air space.

Reflective materials interact directly with electromagnetic waves rather than direct contact or fluid mass movement.

2

Differentiate between the three fundamental mechanisms of thermal energy transfer.

Thermal conduction requires direct solid contact, convection requires fluid circulation, and thermal radiation travels as electromagnetic waves across empty or gas-filled space.

Reflective surfaces reflect incoming infrared radiation and have very low thermal emissivity, preventing radiative heat transfer across the air space.

Key Concept

Thermal Radiation and Reflective Barriers

Estimated Time:1m 0s

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