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
