RATIOS, RATES, AND PROPORTIONS RATIOS REDUCING RATIOS TO LOWEST ...

[Pages:3]RATIOS, RATES, AND PROPORTIONS

RATIOS A ratio is the quotient (comparison) of one quantity divided by another quantity of the same kind (same units).

Examples of units: feet, cents, dollars, cars, minutes, gallons, pounds

a "The ratio of a to b" is written b or a:b. a and b are called the terms of the ratio.

REDUCING RATIOS TO LOWEST (SIMPLEST) TERMS Ratios are fractions; therefore, they can be reduced to lowest terms. However, a ratio that is an improper fraction should not be rewritten as a whole number or as a mixed number. Note that in a ratio the units are NOT written.

RATES A rate is the quotient (comparison) of one quantity divided by a quantity of a different kind (different units).

REDUCING RATES TO LOWEST (SIMPLEST) TERMS Rates are fractions; therefore, they can be reduced to lowest terms. However, a rate that is an improper fraction should not be rewritten as a whole number or as a mixed number. Note that in a rate the units ARE written. Examples of rates:

55 miles

300 miles 20 miles

55 miles per hour = 1 hour , 300 miles on 15 gallons = 15 gallons = 1 gallon ,

3 dollars 3 dollars for 2 hotdogs = 2 hotdogs

PROPORTIONS A proportion is a statement that two ratios or rates are equal to each other.

a= c Consider the proportion b d :

A proportion is said to be true if the fractions are equal when written in lowest terms. In any true proportion, the cross-products are equal. In other words, ad = bc.

Recall, if the cross-products are equal, both fractions are equivalent. Therefore, both fractions can be reduced or built to the same fraction.

SOLVING A PROPORTION FOR n 1. Copy the equation you need to solve. 2. Find the cross-products. 3. Divide both sides of the equation by the coefficient of n. 4. Simplify both sides of the equation. 5. Check the solution by replacing the variable with the value found in Step 4. Then cross-multiply and verify that the cross-products are equal, or verify that both fractions reduce to the same fraction.

Ex: A = n AD = Bn AD = Bn n= AD

BD

BB

B

Ex: A = C An= BC An = BC n= BC

Bn

AA

A

SOLVING AN APPLIED PROBLEM BY USING A PROPORTION 1. Use n (or the variable of your choice) to represent the unknown quantity. Use the given conditions to form two ratios or two rates. 2. Form a proportion by setting the two ratios or two rates equal to each other. Be sure that the units occupy corresponding positions in the proportion (see Ex:). 3. Drop the units. 4. Solve the proportion for n.

miles A = miles B - OR - miles A = hoursA

Ex: hoursA hoursB

miles B hoursB

USING UNIT FRACTIONS AND RATES TO SOLVE APPLIED PROBLEMS A unit fraction is a fraction whose value is 1. We can convert from one unit of measure to another by multiplying the given measurement by a properly selected unit fraction.

1 foot , 12 inches , 1 hour

4 quarts ,

Exs: 12 inches

1 foot

60 minutes 1 gallon

If the unit to be eliminated is in the numerator, multiply by a unit fraction that has that unit in its denominator. If the unit to be eliminated is in the denominator, multiply by a unit fraction that has that unit in its numerator.

Some applied problems that involve rates can be solved by canceling units.

Ex: Convert "55 miles per hour" to "feet per minute."

55 miles x 5,280 feet x

1 hour

(55)(5,280) =

feet = 4,840 feet per minute

hours

1 mile

60 minutes

60 minute

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download