Q.1. Haretown and Tortoiseville are 54 miles apart. A hare travels at 8 miles
per hour from Haretown to Tortoiseville, while a tortoise travels at 1
miles per hour from Tortoiseville to Haretown.
If both set out at the same time, how many miles will the hare have to travel before meeting the tortoise en route?
Ans: 1.
The hare and the tortoise are together covering the distance at 9 miles per hour (i.e., on adding their speeds).
So, they will cover the distance of 36 miles in 4 hours.
Thus, in 4 hours, they will meet and the hare will have traveled 32 miles.
Alternative Solution through Equations:
Note that : Distance = Speed × Time
Let t be the time before the hare and the tortoise meet.
In t hours, the hare will travel 8 t miles.
In t hours, the tortoise will travel 1 t miles.
Now,
8 t + 1 t = 36
So, t = 36 ⁄ 9 = 4 hours.
Thus, distance traveled by hare before meeting = 8 × 4 = 32 miles.
Q.2. Man Wrinkle spent one-fourth of his life as a boy, one-eighth as a youth, and one-half as an active man. If Man Wrinkle spent 11 years as an old man, then how many years did he spend as an active man?
Ans: 2.
Fraction of life as a boy = 1/4
Fraction of life as a youth = 1/8
Fraction of life as an active man = 1/2
Fraction of life as boy, youth and active man = 1/4 + 1/8 + 1/2 = (2 + 1 + 4)/8 = 7/8
Fraction of life as an old man = 1 − 7/8 = 1/8
Thus, one-eighth of Man Wrinkle's life (as an old man) is 11 years.
So, Man Wrinkle's Age = 88 years.
It may be noted that:
Life as boy = 88/4 = 22 years.
Life as youth = 88/8 = 11 years.
Life as active man = 88/2 = 44 years.
Life as old man = 88/8 = 11 years.
The problem may also be solved by setting up the following equation: a/4 + a/8 + a/2 + 11 = a
where a denotes Man Wrinkle's age in years.
The equation may be solved as shown below.
7a/8 + 11 = a
11 = a − 7a/8 = a/8
a/8 = 11 or a = 88.
Therefore, life spent as active man = 88/2 = 44 years.
Q.3.Dad gives you money every day to put in your new piggy bank. He gives money to you in such a way that the money in the piggy bank doubles with each passing day.
If you already have 1 cent in the piggy bank and Dad gives you 1 cent the first day, 2 cents the second day, 4 cents the third day and so on, then your piggy bank gets full on the 15th day.
1. On which day will your piggy bank be half-full? For example, type 6 if your answer is 6th day.
2. In addition to Dad's contributions, if Mom also gave you 1 cent the first day, 2 cents the second day, 4 cents the third day and so on, then on which day would your piggy bank be about half-full?
For example, type 6 if your answer is 6th day.
Ans: 3.
1. Since the money in the piggy bank doubles with each passing day, the piggy bank will be half-full the day previous to the one on which it gets full. Thus, the piggy bank will be half-full on the 12th day.
2. If both Mom and Dad contribute equal amounts to your piggy bank, then each needs to only make your piggy bank quarter-full. When both Mom and Dad contribute, the piggy bank will be half-full two days prior to the day it would be full when only Dad contributes. Thus, the piggy bank will be full on the 11th day when both Mom and Dad contribute.
If both set out at the same time, how many miles will the hare have to travel before meeting the tortoise en route?
Ans: 1.
The hare and the tortoise are together covering the distance at 9 miles per hour (i.e., on adding their speeds).
So, they will cover the distance of 36 miles in 4 hours.
Thus, in 4 hours, they will meet and the hare will have traveled 32 miles.
Alternative Solution through Equations:
Note that : Distance = Speed × Time
Let t be the time before the hare and the tortoise meet.
In t hours, the hare will travel 8 t miles.
In t hours, the tortoise will travel 1 t miles.
Now,
8 t + 1 t = 36
So, t = 36 ⁄ 9 = 4 hours.
Thus, distance traveled by hare before meeting = 8 × 4 = 32 miles.
Q.2. Man Wrinkle spent one-fourth of his life as a boy, one-eighth as a youth, and one-half as an active man. If Man Wrinkle spent 11 years as an old man, then how many years did he spend as an active man?
Ans: 2.
Fraction of life as a boy = 1/4
Fraction of life as a youth = 1/8
Fraction of life as an active man = 1/2
Fraction of life as boy, youth and active man = 1/4 + 1/8 + 1/2 = (2 + 1 + 4)/8 = 7/8
Fraction of life as an old man = 1 − 7/8 = 1/8
Thus, one-eighth of Man Wrinkle's life (as an old man) is 11 years.
So, Man Wrinkle's Age = 88 years.
It may be noted that:
Life as boy = 88/4 = 22 years.
Life as youth = 88/8 = 11 years.
Life as active man = 88/2 = 44 years.
Life as old man = 88/8 = 11 years.
The problem may also be solved by setting up the following equation: a/4 + a/8 + a/2 + 11 = a
where a denotes Man Wrinkle's age in years.
The equation may be solved as shown below.
7a/8 + 11 = a
11 = a − 7a/8 = a/8
a/8 = 11 or a = 88.
Therefore, life spent as active man = 88/2 = 44 years.
Q.3.Dad gives you money every day to put in your new piggy bank. He gives money to you in such a way that the money in the piggy bank doubles with each passing day.
If you already have 1 cent in the piggy bank and Dad gives you 1 cent the first day, 2 cents the second day, 4 cents the third day and so on, then your piggy bank gets full on the 15th day.
1. On which day will your piggy bank be half-full? For example, type 6 if your answer is 6th day.
2. In addition to Dad's contributions, if Mom also gave you 1 cent the first day, 2 cents the second day, 4 cents the third day and so on, then on which day would your piggy bank be about half-full?
For example, type 6 if your answer is 6th day.
Ans: 3.
1. Since the money in the piggy bank doubles with each passing day, the piggy bank will be half-full the day previous to the one on which it gets full. Thus, the piggy bank will be half-full on the 12th day.
2. If both Mom and Dad contribute equal amounts to your piggy bank, then each needs to only make your piggy bank quarter-full. When both Mom and Dad contribute, the piggy bank will be half-full two days prior to the day it would be full when only Dad contributes. Thus, the piggy bank will be full on the 11th day when both Mom and Dad contribute.
