RATIOS, PROPORTIONS, AND RATES

1. Setting up a Ratio

To find a ratio, put the number associated with the word of in the nominator and the quantity associated with the word to in the denominator. Then reduce.   The ratio of  15 cakes to 12 candys is 15/12,  which reduces to 5/4.

2. Part-to-Part Ratios and Part-to-Whole Ratios

If the parts add up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number in the original ratio over the sum of the numbers.

Example:  If the ratio of cats to dogs is 1 to 5, then the cat-to-whole ratio is 1 / (1 + 5) = 1/6

and the dog-to-whole ratio is 5 / (1 + 5) = 5/6.  In other words, 5/6 of the animals are dogs.

3. Using Ratios to Solve Rate Problems

Example: If snow is falling at the rate of one foot every four hours, how many inches of snow will fall in seven hours?

Setup:

1 foot =       x inches

4 hours                         7 hours

Make the units the same:

12 inches =    x inches

4 hours             7 hours

Solve:

4x= 12 X 7

x= 21

4. Average Rate

Average rate is NOT simply the average of the rates.

Total A

Average A per B =         Total B

Total distance

Average Speed =          Total time

To find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, don't just average the two speeds.   First figure out the total distance and the total time. The total distance is 120 + 120 = 240 miles. The times are two hours for the first leg and three hours for the second leg, or five hours total. The average speed, then, is 240/5 = 48 miles per hour.

5)   Common Formulas for Word Problems:

a)  Distance = Rate x Time

Example:  Two cars leave Miami at the same time traveling in opposite directions.  One car travels at 60 mph and the other travels at 50 mph.  In how many hours will they be 880 miles apart?

Let R1 be the rate of the first car;  let R2 be the rate of the second car

Let T1 be the time of the first car;  let T2 be the time of the second car

The distance the first car travels is R1 x T1 and the distance the second car travels is R2 x T2

R1 T1 + R2 T2 = 880.  We also know that T1 = T2.  Our new equation is:

60T + 50T = 880

T = 8

It will take 8 hours for the cars to be 880 miles apart.

b)  Work = Rate x Time

Example:  If Jasmine can sew a dress alone in 6 days and Amy can sew the same dress in 8 days, how long will it take them to sew the dress if they both work on it?

Let x be the number of hours if they work together.

Jasmine                        Amy                 Together

Hours to sew                             6                                  8                      x

Part done in one day                 1                                  1                      1

1/6  +  1/8  =  1/x

Solving for x, we get 3  3/7 days

c)  Interest = Principal Amount x Rate x Time

Example:  If Michelle has $6,700 in a bank that pays 4% simple interest for three years, how much interest will she earn in three years?  (Assume no compounding).

Interest = Principal Amount x Rate x Time

Interest = (6700)(0.04)(3) = $804