Algebra - Converting Measurements

A map is drawn with a scale factor of a mile per quarter inch. If a location is 3.75 inches away, how many miles is this?

$15\textup{ miles }$

$12\textup{ miles }$

$13.5 \textup{ miles }$

$15.5 \textup{ miles }$

$17.25 \textup{ miles }$

Explanation

Write the conversion relating miles and inches. The map is drawn where one mile is traveled per quarter inch marked.

$(\frac{1 \textup{ mile}}{\frac{1}{4}\textup{ inch}})$

Rewrite 3.75 into a fraction.

$3.75 \textup{ inches } = 3\frac{3}{4}= \frac{15}{4}\textup{ inches}$

Multiply this dimension with the conversion to determine how many miles is the certain location.

$\frac{15}{4}\textup{ inches}(\frac{1 \textup{ mile}}{\frac{1}{4}\textup{ inch}})$

The inches unit will cancel, and will leave us with the mile unit. Multiply the numerator with the numerator and the denominator with the denominator.

$\frac{15}{4}\textup{ inches}(\frac{1 \textup{ mile}}{\frac{1}{4}\textup{ inch}}) = 15\textup{ miles }$

The location is 15 miles away.

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

$10 \frac{8}{9} \textrm{ ft}^{2}$

$8 \frac{1}{9} \textrm{ ft}^{2}$

$9 \frac{8}{9} \textrm{ ft}^{2}$

$9 \frac{1}{9} \textrm{ ft}^{2}$

$9 \frac{4}{9} \textrm{ ft}^{2}$

Explanation

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:

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

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?

Explanation

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.

Convert two days into seconds.

172,800 seconds

86,400 seconds

2,880 seconds

None of these answers

187,200 seconds

Explanation

To convert from one measurement to another simply multiply or divide by all the measurements in between. Notice how each successive fraction has a denominator that cancels with the previous fraction so that we are finally left with just the measurement we want. Be careful to make sure the that you always write a correct ratio.

$48\hspace{1mm}hours*\frac{60\hspace{1mm}minutes}{1\hspace{1mm}hour}*\frac{60\hspace{1mm}seconds}{1\hspace{1mm}minute}=172,800\hspace{1mm}seconds$

How many days are in seconds?

Explanation

Recall that there are seconds in a minute, minutes in an hour, and hours in a day.

$691,200\text{ seconds}\times\frac{1\text{ minute}}{60\text{ seconds}}\times\frac{1\text{ hour}}{60\text{ minutes}}\times\frac{1\text{ day}}{24 \text{ hours}}=8 \text{ days}$

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?

$1,000X ; \textrm{kg}$

$\frac{1,000}{X} ; \textrm{kg}$

$X ; \textrm{kg}$

$1,000,000X ; \textrm{kg}$

$\frac{1,000,000}{X} ; \textrm{kg}$

Explanation

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:

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

Convert to .

Explanation

In order to complete this conversion we have to know that there are per . So this problem can be solved by dividing our by , or moving the decimal place to the left by .

$\frac{74L}{1}\cdot\frac{1kL}{1000L}=0.074kL$

Convert to .

Explanation

In order to complete this conversion we have to know that there are per . So this problem can be solved by multiplying our by , or moving the decimal place to the right by .

$\frac{20L}{1}\cdot\frac{1000mL}{1L}=20,000mL$

Pecans can be purchased in bulk for $8.50 per pound at Charlie's grocery; the weight of the pecans purchased need not be a whole number of pounds. David has dollars on hand and wants to purchase as many pecans as he can afford. In terms of , how many ounces can he purchase?