Linear Equations - Algebra 1

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Question

Harold has a (full) two liter bottle of soda in his refrigerator. He takes it out and pours eight-ounce glasses of soda for himself and each of his three friends that have come over. How many liters of soda are left in the bottle (rounded to two decimal places)?

1 liter = 33.814 fluid ounces

Answer

Convert the two liters of soda into ounces.

Subtract the four eight-ounce glasses of soda that were poured.

Convert the remaining ounces back into liters.

$35.628 * \frac{1}{33.814} = 1.05$

Alternatively, you could set it up as one equation and solve for , as shown below.

$x = 2 - [(8\times 4)\times\tfrac{1}{33.814}]$

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Question

Convert 60 mph (miles per hour) into fps (feet per second).

Answer

To replace hours with seconds, you must divide by 3600 (seconds in an hour), and to replace miles with feet you must multiply by 5280 (number of feet in a mile).

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Question

Albert is traveling at 70 kilometers per hour, Joseph is moving at 35 meters per second, and Katrina is moving at 1.9 kilometers per minute. Which person is traveling the fastest?

Answer

To compare these three different measurements, we need to convert them so they have the same units. Let's convert all three values into meters per second.

Multiplying 70 km/h by 1000 m/km gives us 70,000 meters per hour, and dividing that value by 3600 (since there are 3600 seconds in an hour) gives us 19.4. So, Albert is traveling at 19.4 meters per second.

We convert Katrina's speed in a similar way, multiplying 1.9 km/minute by 1000 m/km and then dividing by 60 (since there are 60 seconds in a minute). These calculations give us 31.67 m/s.

Since both of these values are slower than Joseph's speed of 35 m/s, Joseph is traveling the fastest.

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Question

Edward is driving miles to go home for Christmas vacation. The last time Edward took this trip, he drove miles per hour. How much shorter would Edward's trip be this time if he drives miles per hour?

Answer

This problem relies on the formula $\textup{distance = rate * time}$ , or .

Solving for in the case of both Edward's original trip and the trip Edward will take this time will give us the time each trip would take. The difference between these times is the answer.

The distance Edward travels is miles. The rate at which he drives on the first trip is miles per hour. Remember that in writing out equations we can treat the word "per" as a division sign, so we express $60 \textup{ miles per hour}$ as $\frac{60 \ miles}{1 \ hour}$ . Solve for with these numbers in .

$225 \ miles = (\frac{60 \ miles}{1 \ hour}) * (t \ hours)$

$(225 \ miles) * (\frac{1 \ hour}{60 \ miles}) = (\frac{60 \ miles}{1 \ hour}) * (t \ hours) * (\frac{1 \ hour}{60 \ miles})$

$\frac{225}{60} \ hours = t \ hours$

$t = \frac{15}{4} \ hours = 3\tfrac{3}{4} \ hours$

To convert the remainder fraction of an hour into minutes, multiply it by $60 \ minutes \ per \ hour$ .

$\frac{3}{4} \ hours * \frac{60 \ minutes}{1 \ hour} = \frac{180}{4} \ minutes = 45 \ minutes$

So the total time for the original trip is hours and minutes.

Now use the same equation, but replace the miles per hour with miles per hour for the second trip.

$225 \ miles = (\frac{75 \ miles}{1 \ hour}) * (t \ hours)$

$(225 \ miles) * (\frac{1 \ hour}{75 \ miles}) = (\frac{75 \ miles}{1 \ hour}) * (t \ hours) * (\frac{1 \ hour}{75 \ miles})$

$\frac{225}{75} \ hours = t \ hours$

$t = 3 \ hours$

We now know the trip at miles per hour would be hours long. Subtracting this from the length of the original miles per hour trip gives us the amount of time by which this trip would be shorter than the original.

$(3 \ hours, 45 \ minutes) - (3 \ hours) = 45 \ minutes$

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Question

Daycare currently costs $240 per week. Starting next year, there will be a 15% increase in the cost of daycare. How much will one month of daycare cost?

Assume one month equals four weeks.

Answer

15% is equal to $\small \frac{15}{100}$ , or 0.15. We can find the amount of the increase by multiplying the original cost by 0.15.

The new amount is the sum of the original amount and the increase.

$\small 240+36=276$

This gives us the new cost for one week. The cost for the new month will be four times this amount.

$\small 276*4=1104$

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Question

On a faraway planet, the measurement of distance has been warped, so that one foot contains only eight inches, instead of the twelve inches it takes to equal one foot here on Earth. On this faraway planet, how many inches are in 3.5 feet?

Answer

Use a proportion to answer this question, keeping in mind that one foot is eight inches on this faraway planet.

$\frac{1 ft}{8 inches} = \frac{3.5ft}{x}$

Cross-multiply and simplify.

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Question

What is the value of 4.5 meters per second in kilometers per hour?

Answer

This conversion has two components: converting meters into kilometers and converting seconds into hours. Let's focus on seconds first. Because there are 3600 seconds in one hour, we need to multiply $\frac{4.5\ m}{1\ s}\cdot \frac{3600\ s}{1\ hr}$

giving us

$\frac{16,200\ m}{hr}$

Next, since there are 1000 meters in one kilometer, we multiply $\frac{16,200\ m}{1\ hr}\cdot \frac{1\ km}{1000\ m}$

giving us a final answer of 16.2.

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Question

The density of a substance is grams per cubic centimeter. In terms of , what is the mass, in kilograms, of one cubic meter of this substance?

Answer

1 meter is equal to 100 centimeters, so 1 cubic meter is equal to $100^{3} = 1,000,000$ cubic centimeters.

1,000,000 cubic centimeters has mass:

$1,000,000 \textrm{;cm}^{3} \cdot X \frac{\textrm{g}}{\textrm{cm}^{3}} = 1,000,000X \textrm{; g}$

1,000 grams are in a kilogram, so convert to kilograms by dividing this number by 1,000:

$1,000,000 X \div 1,000 = 1,000 X$

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Question

Three centimeters on a map represents 45 actual kilometers. In terms of , how many centimeters on the map represent actual kilometers?

Answer

Let be the number of map centimeters that represent kilometers.

Set up a proportion statement which equates two ratios, each of which compares map centimeters to actual kilometers represented. Then solve for

$\frac{x}{N} = \frac{3}{45}$

$\frac{x}{N} = \frac{1}{15}$

$\frac{x}{N} \cdot N = \frac{1}{15}\cdot N$

$x = \frac{N}{15}$

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Question

Three centimeters on a map represents 66 actual kilometers. In terms of , how many centimeters on the map represent actual kilometers?

Answer

Let be the number of map centimeters that represent kilometers.

Set up a proportion statement which equates two ratios, each of which compares map centimeters to actual kilometers represented. Then solve for

$\frac{x}{N} = \frac{3}{66}$

$\frac{x}{N} = \frac{1}{22}$

$\frac{x}{N} \cdot N = \frac{1}{22}\cdot N$

$x = \frac{N}{22}$

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Question

Three centimeters on a map represents 66 actual kilometers. In terms of , how many actual kilometers are represented by centimeters on the map?

Answer

Let be the number of actual kilometers represented by map centimeters.

Set up a proportion statement which equates two ratios, each of which compares actual kilometers represented to map centimeters. Then solve for .

$\frac{x}{N} = \frac{66}{3}$

$\frac{x}{N} = 22$

$\frac{x}{N} \cdot N = 22 \cdot N$

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Question

Three centimeters on a map represents 45 actual kilometers. In terms of , how many actual kilometers are represented by centimeters on the map?

Answer

Let be the number of actual kilometers represented by map centimeters.

Set up a proportion statement which equates two ratios, each of which compares actual kilometers represented to map centimeters. Then solve for .

$\frac{x}{N} = \frac{45}{3}$

$\frac{x}{N} = 15$

$\frac{x}{N} \cdot N = 15 \cdot N$

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Question

1 mile = 5,280 feet

Kevin ran 10,000 feet on his 30-minute morning run. If he kept up that pace, how many miles would Kevin run in 1 hour? (to two decimal places)

Answer

Set up an equation that says 10,000 feet per 30 minutes equals (unknown) miles in one hour, like below.

$\frac{10,000}{30}=\frac{x}{1}$

Multiply the left side of the equation by the conversion rates to change feet/minute to miles/hour.

$\frac{10,000}{30}\times\frac{1}{5280}\times\frac{60}{1}=\frac{600,000}{158,400}=3.79$

Check if this answer makes sense. There are 5,280 feet in a mile so 10,000 feet is a bit less than two miles. 30 minutes is half an hour so 20,000 should be a bit less than four miles.

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Question

Convert 42.6 inches to meters:

1 in = 2.54 cm

1 m = 100 cm

Answer

$\frac{42.6in}{1}\times \frac{2.54cm}{1in}\times\frac{1m}{100cm}=1.08m$

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Question

A rectangle has dimensions 56 inches by 28 inches. Write its area in square feet.

Answer

12 inches make a foot, so we can convert the dimensions as follows:

$56 \textrm{ in }\div \frac{12 \textrm{ in }}{1 \textrm{ ft }} = \frac{56}{12} \textrm{ ft }$ ;

$\frac{56}{12} = \frac{56 \div 4}{12 \div 4}= \frac{14}{3} \textrm{ ft }$

$28 \textrm{ in }\div \frac{12 \textrm{ in }}{1 \textrm{ ft }} = \frac{28}{12} \textrm{ ft }$ ;

$\frac{28}{12} = \frac{28 \div 4}{12 \div 4}= \frac{7}{3} \textrm{ ft }$

Now multiply these dimensions to get the area in square feet:

$\frac{14}{3} \cdot \frac{7}{3} = \frac{98}{9} = 10 \frac{8}{9} \textrm{ ft}^{2}$

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Question

Convert three yards to inches.

Answer

To solve this problem, we need to know the conversions between yards to feet, and feet to inches. Write their correct conversions.

$1 \textup{ yard}= 3\textup{ feet}$

$1\textup{ foot}= 12 \textup{ inches}$

Convert three yards to feet.

$3 \textup{ yards}\left(\frac{3 \textup{ feet}}{1 \textup{ yard}}\right)= 9 \textup{ feet}$

Convert nine feet to inches.

$9 \textup{ feet}\left(\frac{12 \textup{ inches}}{1 \textup{ foot}}\right)= 108\textup{ inches}$

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Question

Convert one inch to yards.

Answer

Write the conversion factors.

$12\textup{ inch }= 1\textup{ feet}$

$1\textup{ yard}=3 \textup{ feet}$

From here we multiply the conversion factors together making sure the units from the top cancel with those on the bottom to leave yards.

$1\textup{ inch}\left(\frac{1\textup{ feet}}{12\textup{ inch }}\right)\left(\frac{1\textup{ yard}}{3 \textup{ feet}}\right)=\frac{1}{36}\textup{ yards}$

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Question

How many quarts are in $\frac{5}{8}$ of a gallon?

Answer

Four quarts make up a gallon. Write the dimensional analysis.

$4\textup{ quarts }= 1\textup{ gallon}$

Multiply the amount of gallons given in the question by the conversion factor. Make sure to have the units on the top cancel those in the bottom to leave the amount quarts.

$\left(\frac{5}{8}\textup{ gallons }\right)\left(\frac{4\textup{ quarts }}{1\textup{ gallon}}\right) = \frac{5}{2}\textup{ quarts }$

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Question

Convert: $\frac{1}{5} \textup{ feet}$ to inches.

Answer

Write the conversion factor of inches and feet.

$12 \textup{ inches}= 1 \textup{ feet}$

Now use this coversion and multiply it by what was given in the original question.

$\frac{1}{5} \textup{ feet}\left(\frac{12 \textup{ inches}}{1 \textup{ feet}}\right)= \frac{12}{5} \textup{ inches}$

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Question

Convert feet to inches.

Answer

We know inches make up foot. Since we are looking for inches in feet, we can write a proportion.

$\frac{1\ foot}{12\ inches}=\frac{2\ feet}{x\ inches}$

Cross-multiply.

$(x\ inches)(1\ foot)=(2\ feet)(12\ inches)$

Divide each side by foot.

$x\ inches= 2(12\ inches)$

$x=24\ inches$

Therefore, $2\ feet = 24\ inches$ .

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