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CHAPTER 1
Proportional Reasoning: An Overview
GETTING STARTED
1. If 6 men can build a house in 3 days, then to shorten the time, you will need more men on the job. As the number of men goes up, the number of days goes down. Six men build
of the house in one day, so 12 men could build
of the house in one day, and 18 men could build
of one whole house in a day (assuming that all of the men do an honest day's work). This means that it would take 3 times as many men to complete the job in
the time.
2. If 6 chocolates cost $0.93, then 12 would cost $1.86 and 24 would cost $3.72. If 6 cost $0.93, then 2 cost $0.31. The cost of 22 candies is $0.31 less than $3.72 or $3.41.
3. John has 3 times as many marbles as Mark, so you can think of the whole set of marbles as 4 equal groups, with 3 of the groups in front of John and 1 group in front of Mark. If there are 32 marbles, each group contains 8 marbles. John has 24 marbles and Mark has 8 marbles.
4. If Mac does twice as much as his brother, he will do
of the lawn, while his brother does
. If it takes Mac 45 minutes to do
of the job, then it takes 15 minutes to do
and 30 minutes to do
. Meanwhile, during that 30 minutes, his little brother does the other
of the lawn.
5. The more people you have working, the faster the job will get done (assuming, of course, that the boys do not goof off on the job). If 6 boys were given 20 minutes to clean up, then 1 boy should be given 6 times as much time or 120 minutes. Then 9 would each require
of the time needed by 1 person:
minutes. Another way. If 6 boys can do the job in 20 minutes, then 3 would take
40 minutes. If there are 9 people working, then each set of 3 people does
of the job in 40 minutes, so the total time needed is
minutes.
6. No answer. Knowing the weight of one player is not helpful in determining the weight of 11 players. People's weights are not related to each other.
7. Every time they put away $7, Sandra pays $2 and her mom pays $5. To get $210, they will need to make their respective contributions 30 times. In all, Sandra will contribute $60 and her mom will contribute $150.
8. If you decrease the number of people doing a job, it will take longer to finish the job. If you have
the number of people working, they can get only
as much done in the 96 minutes. Each
of the job will take them 96 minutes, so they can do
of work in 192 minutes and
of the job (the whole job) in 288 minutes (4 hours and 48 minutes).
9. Reason down then reason up. The bike can run for 5 minutes on $0.65 worth of fuel, and for 1 minute on $0.13 worth of fuel. It can run for 6 minutes on $0.78 worth of fuel and for 7 minutes on $0.91 worth of fuel.
10. 15 to 1 = 150 to 10. Therefore, decreasing the number of faculty by 8 will give the required ratio.
11. Your shadow (8′) is 1.6 times as tall as you are (5′), so the shadow cast by the telephone pole must be 1.6 times as tall as the pole. If the shadow is 48′ and that is 1.6 times the real height of the telephone pole, the pole must be 30′ tall. Also, the telephone pole's shadow is 6 times as long as yours, so it must be 6 times as tall as you are.
12. In a square, two adjacent sides have the same length. That means that the ratio of the measure of one side to the measure of the other side would be 1. The rectangle that is most square is the one whose ratio of width to length is closest to 1. For the 35″ × 39″ rectangle, the ratio is
or about 0.90; for the 22″ × 25″ rectangle, the ratio is
or 0.88. This means that the 35″ × 39″ rectangle is most square.
13. Gear A has 1.5 times the number of teeth on gear B. So every time A turns once...
