CHAPTER 4

Question 1

26 PROBLEMS DIFFERENTIAL EQUATION

To stimulate his son in the pursuit of partial differential equations, a math professor offered to pay him $8 for every equation correctly solved and to fine him $5 for every incorrect solution. At the end of 26 problems, neither owed any money to the other. How many did the boy solve correctly?

Answer 1

10

simple algebra

Question 2

EXPERT GOES TO RACES

An expert on transformer design relaxed one Saturday by going to the races. At the end of the first race he had doubled his money. He bet $30 on the second race and tripled his money. He bet $54 on the third race and quadrupled his money. He bet $72 on the fourth race and lost it, but still had $48 left. With how much money did he start?

Answer 2

$29

Question 3

MATHEMATICIAN BROKEN CLOCK? TIME EQUATION

A mathematician whose clock had stopped wound it, but did not bother to set it correctly. Then he walked from his home to the home of a friend for an evening of hi-fi music. Afterwards, he walkedback to his own home and set his clock exactly. How could he do this without knowing the time his trip took?

equation?

Answer 3

correct time:
t2 + 1/2 [(T2-T1)-(t2-t1)]

where:
T1 = setting of his clock when he left home
T2 = setting of his clock when he returned
T2-T1 : total length of time he was away

t1 = correct time he arrived at his
friend’s home
t2 = correct time he left his friend’s home
t2-t1 = length of time at his friend’s home

Question 4

SING SING, TARRY TOWN

Between Sing-Sing and Tarry-Town,
I met my worthy friend, John Brown,
And seven daughters, riding nags,
And every one had seven bags.
In every bag were thirty cats,
And every cat had forty rats,
Besides a brood of fifty kittens.
All but the nags were wearing mittens!
Mittens, kittens - cats, rats - bag, nags - Browns,
How many were met between the towns?

Answer 4

764, 488

Question 5

INFINITE PRODUCT EQUATION

For X < 1 evaluate the infinite product: (1 + X + X^2 + … + X^9) (1 + X^10 + X^20 + X^30 + … + X^90) (1 + X^100 + X^200 + … + X^900)
(…

Answer 5

• from n = 0 to ∞
• ∑ x^n = 1 / (1-x)

Question 6

PRODUCT, CONSTANT?

If v varies as w^2, w^3 as x^4, x^5 as y^6, and y^7 as z^4,
show that the product
v/z * w/z * x/z * y/z
does not vary at all.

Answer 6

their product is a constant

Question 7

THE MINUTE AND THE HOUR HAND ARE EXACTLY TOGETHER EVERY 65 MIN.

Dr. Reed, arriving late at the lab one morning, pulled out his watch and said, “I must have it seen to. I have noticed that the minute and the hour hand are exactly together every sixty five minutes.” Does Dr. Reed’s watch gain or lose, and how much per hour?

Answer 7

In one hour, it gains 60/143 min or 0.42 min.

Question 8

CLOCK BETWEEN 4:00 - 5:00

At this moment, the hands of a clock in the course of normal operation describe a time somewhere between 4:00 and 5:00 on a standard clock face. Within one hour or less, the hands will have exactly exchanged positions; what time is it now?

Answer 8

26.8531 min after 4:00

Question 9

TWO MEN AND A TRAIN

Two men are walking towards each other at the side of a railway. A freight train overtakes one of them in 20 seconds and exactly ten minutes later meets the other man coming in the opposite direction. The train passes this man in 18 seconds. How long after the train has passed the second man will the two men meet? (Constant speeds are to be assumed throughout.)

Answer 9

5562 sec or 1.545 hour

Question 10

FRENCH TRICOLOR

Using the French Tricolor as a model, how many flags are possible with five available colors if two adjacent rows must not be colored the same?

Answer 10

50 flags

Question 11

HEIGHT LADDER TOUCHES THE WALL

A cubic box with sides ‘a’ feet long is placed flat against a wall. A ladder ‘p’ feet long is placed in such a way that it touches the wall as well as the free horizontal edge of the box. If a = 1 and p = √15 calculate at what height the ladder touches the wall, using quadratics only.

Answer 11

3.62ft or 1.38 ft from the floor

Question 12

ESCALATOR PROBLEM

Dr. Irving Weiman, who is always in a hurry, walks up an upgoing escalator at the rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator.

Answer 12

80 steps

Question 13

INCOME TAX AND SALARY IN RUPEES

Citizens of Franistan pay as much income tax (percentage-wise) as they make rupees per week. What is the optimal salary in Franistan?

Answer 13

50 rupees

Question 14

TRIANGULAR FLIGHTS

There are nine cities which are served by two competing airlines. One or the other airline (but not both) has a flight between every pair of cities. What is the minimum number of triangular flights (i.e., trips from A to B to C and back to A on the same airline)?

Answer 14

12 number of triangular flights

Question 15

TWO SNAILS, TWO BITS OF FOOD

Two snails start from the same point in opposite directions toward two bits of food. Each reaches his destination in one hour. If each snail had gone in the direction the other took, the first snail would have reached his food 35 minutes after the second. How do their speeds compare?

Answer 15

V1 = 3/4 V2

first snail travelled 3/4 the speed of the second snail

Question 16

PEARL NECKLACE

A necklace consists of pearls which increase uniformly from a weight of 1 carat for the end pearls to a weight of 100 carats for the middle pearl. If the necklace weighs altogether 1650 carats and the clasp and string together weigh as much (in carats) as the total number of pearls, how many pearls does the necklace?

Answer 16

33 pearls

note:
weight increment = 99/16

Question 17

A pupil wrote on the blackboard a series of fractions having positive integral terms and connected by signs which were either all + or all x , although they were so carelessly written it was impossible to tell which they were. It still wasn’t clear even though he announced the result of the operation at every step. The third fraction had denominator 19. What was the numerator?

Answer 17

1st fracion = 5/3
2nd fracion = 5/2
3rd fracion = 25/19

Question 18

JAI ALAI BALLS,NUMBER?

Jai Alai balls come in boxes of 8 and 15; so that 38 balls (one small box and two large) can be bought without having to break open a box, but not 39. What is the maximum number of balls be bought without breaking boxes?

Answer 18

97 balls

Question 19

TWO EXPRESSION

Without performing any algebraic manipulation at all, Archimedes O’toole remarked that the sum and product of the two expressrions
(x + y - |x-y|) / 2
and
(x + y + |x-y|) / 2
are respectively x+y and
xy. Why was this obvious?

Answer 19

The two expressions are identically equal, respectively, to the smaller and the larger of the two numbers x and y.

Question 20

PARKING LOT, HOW LONG DID EACH PARK?

A parking lot charges X for the first hour or fraction of an hour and 2/3 X for each hour or fraction thereafter. Smith parks 7 times as long as Jones, but pays only 3 times as much. How long did each park? (The time clock registers only in 5-minute intervals.)

Answer 20

Jones = 0.5 hour
Smith = 3.5 hour

Question 21

MR. FIELD, PERCENTAGE IS HE EXCEEDING SPEED LIMIT

Mr. Field, a speeder, travels on a busy highway having the same rate of traffic flow in each direction. Except for Mr. Field, the traffic is moving at the legal speed limit. Mr. Field passes one car for every nine which he meets from the opposite direction. By what percentage is he exceeding the speed limit?

Answer 21

25%

Question 22

THE MILLIONTH TERM

What is the millionth term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,…in which each positive integer n occurs in blocks of n terms?

Answer 22

1414

equation:
k = [1 + √(8n - 7)] /2

Question 23

CORRECT = +1 POINT, BLANK = -1 PONT, INCORRECT = -2 POINT

The teacher marked the quiz on the following basis: one point for each correct answer, one point off for each question left blank and two points off for each question answered incorrectly. Pat made four times as many errors as Mike, but Mike left nine more questions blank. If they both got the same score, how many errors did each make?

Answer 23

Pat = 8 errors
Mike = 2 errors

Question 24

LOGARITHM

A student, just beginning the study of logarithms, was required to evaluate an expression of the form log A / log B. He proceeded to cancel common factors in both numerator and denominator, (including the “factor” log), and arrived at the result 2/3. Surprisingly, this was correct. What were the values of A and B?

Answer 24

A = 9/4
B = 27/8

Question 25

4 TOWNS IN SQUARE CORNERS

There are four towns at the corners of a square. Four motorists set out, each driving to the next (clockwise) town, and each man but the fourth going 8 mi./hr. faster than the car ahead - thus the first car travels 24 mi./hr. faster than the fourth. At the end of one hour the first and third cars are 204, and the second and fourth 212 (beeline) miles apart. How fast is the first car traveling and how far apart are the towns?

Answer 25

V1 = 50 mi / hr
d = 180 miles far between towns

Question 26

MAGIC SQUARE

Represented above is a “magic square” in which the sum of each row, column, or main diagonal is the same. Using nine different integers, produce a “multiplicative” magic square, i.e., one in which the word “product”is substituted for “sum”.

Answer 26

2^n
or
k^n where k > 1

Question 27

INFINITE EXPRESSION

Solve the equation
√(x+√(x+√(x+…))) = √(x√(x√(x…)))
where both members represent infinite expressions

Answer 27

x = 0 or 2

Question 28

DENOMINATORS AND NUMERATORS CONTINUE INDEFINITELY

Which is greater:
3 + 4/[3+4/(3+…)]
or
3 + [3+(3+…)/4]/4

where the respective denominators and numerators continue indefinitely?

Answer 28

numerator and denominator = 4

note:
both expressions are esoteric representations of 4.

Question 29

PENCIL, ERASER, NOTEBOOK PRICE

A pencil, eraser and notebook together cost $1.00. A notebook costs more than two pencils, and three pencils cost more than four erasers. If three erasers cost more than a notebook, how much does each cost?

Answer 29

P = 26
E = 19
N = 55

Question 30

INTERLOCKED HYPERBOLAS

Depicted above are two interlocked hyperbolas. Impossible? You’re right, but can you prove it?

Answer 30

Two hyperbolas can intersect in no more than 4 points

Question 31

(7 + 4√3)^x - 4(2 + √3)^x = -1

Solve for real values of x:
(7 + 4√3)^x - 4(2 + √3)^x = -1

Answer 31

x = 1

Question 32

SACRED NUMBERS

In Puevigi numbers such as 2, 5, 8, 10, etc., that are the sum of two squares, are considered sacred. Prove that the product of any number of sacred numbers is sacred.

Answer 32

Let
M = A^2 + B^2
N = C^2 + D^2
MN : (A^2 + B^2) (C^2 + D^2) = (AC + BD)^2 +
(AD - BC)^2.
Amen.