What is the value of C?

If digits A,B, and C are added, the sum is the two-digit number AB as shown at the right. What is the value of C?

Given A+B+C = AB
AB means 10th digit is A and B is the units digits, AB is 10A+B

i.e; A+B+C = 10A+B
A+C = 10A
C = 9A

As C is a single digit number, A can be only 1.
If A = 1, C = 9.

Hence C is 9.

What is the smallest number of children the class could have?

If a kindergarten teacher places her children 4 on each bench, there will be 3 children who will not have a place. However if 5 children are placed on each bench, there will be 2 empty places. What is the smallest number of children the class could have?

Let N be the number of children in the class.

If 4 children are placed on each bench, there will be 3 children left out.
=> N-3 is divisible by 4

If 5 children are placed on each bench, there will be 2 empty seats left out.
i.e. 3 children in one bench.
=> N-3 divisible by 5.

Hence N-3 should be divisible by 4 and 5 and smallest number divisible by 4 and 5 is 20.
=> N-3 = 20
=>N = 23

Smallest number of children the class could have is 23.

What is the perimeter of Figure B?

The small boxes in Figures A and B at the right are congruent squares. The perimeter of Figure A is 48 inches. What is the perimeter of Figure B? (The perimeter of a figure is the distance around it.) 

Let S be the side of the square in inches.

Perimeter of Figure A = 5S+5S+1S+5S = 16S = 48 inches

One side of the square S = 48/3 = 16 inches

Perimeter of Figure B = 3S+3S+3S+4S+3S+4S = 20S = 20 *3 = 60

Perimeter of Figure B is 60 inches.

How much does one loaf of bread cost?

One loaf of bread and six rolls cost $1.80. At the same prices, two loaves of bread and four rolls cost $2.40. How much does one loaf of bread cost?

Let B represent the cost of one loaf of bread and R represent the cost of one roll.

1B+6R=$1.80 = 180 cents ------(1)
2B+4R=$2.40 = 240 cents ------(2)

(1) * 2 gives
2B+12R=360 ----(3)

(3)-(2) gives
12R-4R=360-240
8R=120
R=15

B=180-6R=180-6*15=180-90=90

One loaf of bread costs 90 cents.

Which is greater, D or N, and by how much?

D is the sum of the odd numbers from 1 through 99 inclusive, and N is the sum of the even numbers from 2 through 98 inclusive:
           D = 1+3+5+ ........+99 and N = 2+4+6+ ...... +98
Which is greater, D or N, and by how much?

MetroEye: Refer Progression03 for formulae.

Number of terms in odd number series = \(\dfrac{t_n+1}{2}\)= \(\dfrac{99+1}{2}\) = \(\dfrac{100}{2}\)  = 50

Number of terms in even number series = \(\dfrac{t_n}{2} = \dfrac{98}{2}\)  = 49

Sum, D = \( n^2\) = 50*50

Sum, N = n(n+1) = 49*50

=> D is greater than N.

Difference between D and N = D-N = 50*50-49*59 = 50(50-49) = 50

Hence D is greater than N by 50.

How many people returned on the stagecoach?

A group of 21 people went to the county fair with 9 people on a stagecoach and 3 people in each buggy. On the return trip, 4 people rode in each buggy. How many people returned on the stagecoach?

To trip
Total people in the group = 21
People went on the stagecoach = 9
People went on the buggy = 21-9 = 12
As 3 people went in each buggy, number of buggy = 12/3 = 4

Return trip
As 4 people rode in each buggy, total number of people = 4*4 = 16
People rode on the stagecoach = 21-16 = 5

5 people returned on the stagecoach.

How old is the man?

The age of a man is the same as his wife's age with the digits reversed. The sum of their ages is 99 and the man is 9 years older than his wife. How old is the man?

Let M be the age of the man and W be the age of his wife.

Given sum of their ages is 99 => M+W = 99 ----(1)
Also the man is 9 years older than his wife => M = W+9 ---(2)
i.e; M-W=9 ----(3)

Adding (1) and (3)
2M = 99+9 = 108
M = 54
W = 54-9 = 45

The man is 54 years old.

What day of the week is the first day of the month?

The last Friday of a particular month is on the 25th day of the month. What day of the week is the first day of the month?

Counting back from 25th day, the multiple of 7 is 21'st day which will also be a Friday. i.e; 25-21= 4th day of the month.

If 4th day is a Friday, then first day of month will be Tuesday.

The first day of the month is a Tuesday.

In what digit will a multiplication string of thirty-five 3s end?

3x3, 3x3x3, and 3x3x3x3 are multiplication strings of two 3s, three 3s, and four 3s respectively.  When each string multiplication is done, 3x3 ends in 9, 3x3x3 ends in 7, 3x3x3x3 ends in 1. In what digit will a multiplication string of thirty-five 3s end?

Considering the pattern of multiplication strings and end-digits:

String	End-Digit
3	3
3x3	9
3x3x3	7
3x3x3x3	1
3x3x3x3x3	3
3x3x3x3x3x3	9

After four 3s, the end-digit pattern repeats itself.
=> string of thirty-five 3s means 35/4 = 8*4+3
=> after 8 four group patterns third iteration which is 7.

A multiplication string of thirty-five 3s will end in 7.

How many gallons does a full tank hold?

If 24 gallons of water are poured into an empty tank, then 3/4 of the tank is filled. How many gallons does a full tank hold?

Let V be the full volume of tank.
3/4 of tank = 24 gallons
3/4 * V = 24
V = 24 * 4/3 = 32

The full tank will hold 32 gallons of water.

What is the cost of the purchase?

The number of two-dollar bills I need to pay for a purchase is 9 more than the number of five-dollar bills I need to pay for the same purchase. What is the cost of the purchase?

Let C be the cost of the purchase and let T be the number of two-dollar bills and F be the number of five-dollar bills I have to pay.

C = 5F = 2T = 2(9+F)
5F = 18 + 2F
5F - 2F = 18
3F = 18
F = 6
C = 5F = 5*6 = 30

Cost of the purchase is $30.

After how many stops will the train be full?

A train can hold 78 passengers. The train starts out empty and picks up 1 passenger at first stop, 2 passengers at the second stop, 3 passengers at the third stop, and so forth. After how many stops will the train be full?

MetroEye: Refer to Preogression02

The sequence in which passengers are picked up is 1,2,3.....
So when the sum of consecutive numbers reach 78, the train will be full.

i.e; 1+2+3+......+N = 78

Here S = 78

N = \(\dfrac{\sqrt{1+8S} -1}{2}\) = \(\dfrac{\sqrt{1+8*78} -1}{2}\) = \(\dfrac{\sqrt{1+624} -1}{2}\) = \(\dfrac{\sqrt{625} -1}{2}\) = \(\dfrac{25-1}{2}\) = 12

=> 1+2+3+4+5+6+7+8+9+10+11+12 = 78

So the train will be full after 12 stops.

How long will it take to fill the pool?

Three water pipes are used to fill a swimming pool. The first pipe alone takes 8 hours to fill the pool, the second pipe alone takes 12 hours to fill the pool, and the third pipe alone takes 24 hours to fill the pool. If all three pipes are opened at the same time, how long will it take to fill the pool?

MetroEye: This problem is similar to work-time problem. Refer MetroEye02

As per the formula,
\(\dfrac{1}{t} = \dfrac{1}{8} + \dfrac{1}{12}+ \dfrac{1}{24}= \dfrac{3 + 2 + 1}{24}  =  \dfrac{6}{24} =  \dfrac{1}{4}\)

=> t = 4

It will take 4 hours to fill the pool if all three pipes are opened at the same time.

What value should N have?

In the magic square at the right, the four numbers in each column, in each row, and in each of the two diagonals, have the same sum. What value should N have?

For determining the value of N, we need to find the values A,B and C as in the below figure.

Sum of the columns = 12+9+5+8
                                 = 34

7+9+16+A = 34
A = 34-32 = 2

8+11+A+B = 34
B = 34-8-11-2 = 13


12+C+3+B = 34
C = 34-12-3-13 = 6

N+4+9+C = 34
N = 34-4-9-6 = 15

Value of N is 15.

Under which column letter will 101 appear?

The counting numbers are arranged in four columns as shown at the right. Under which column letter will 101 appear?

Observing the pattern of numbers, arranged into 4 columns - First four numbers are in from left to right and the next four numbers are in reverse order. => After 8 numbers the pattern repeats.

The multiple of 8 nearest to 101 is 101/8 ~ 12 => 12*8 = 96 is the nearest multiple.
So 96 will be in Column A and then 101 - 96 = 5 columns to reach 101. So from A fifth column will be A, B, C, D and then again D.

101 will be under column letter D

How many correct answers did she have?

In a math contest of 10 problems, 5 points was given for each correct answer and 2 points deducted for each incorrect answer. If Nancy answered all 10 problems and scored 29 points, how many correct answers did she have?

Let C be the number of correct answers and I be the number of incorrect answers.

Given total problems is 10 => C+I = 10  -----(1)

Nancy scored 29 points for all 10 problems;
=> 5C-2I = 29 -------(2)

Multiplying equation (1) by 2 gives
2C+2I = 20 -----(3)

(2)+(3) gives
7C = 49
C = 7

Nancy have 7 correct answers.

How many rectangles with different shapes?

The perimeter of a rectangle is 20 feet and the foot-measure of each side is a whole number. How many rectangles with different shapes satisfy these conditions?

The perimeter of a rectangle = 2 (l+b) = 20 feet
=> l+b = 10 feet.

As the sides of the rectangle are whole numbers, the possible values for the sides based on the above condition are 1 by 9, 2 by 8, 3 by 7, 4 by 6, and 5 by 5.

There are 5 different rectangles for the given conditions.

Who had the least amount of money, and how much was it?

When Anne, Betty and Cynthia compared the amount of money each had they discovered that Anne and Betty together had $12, Betty and Cynthia together had $18, and Anne and Cynthia together had $10. Who had the least amount of money, and how much was it?

Let A, B and C represent the amount of money and Anne, Betty and Cynthia has.

A+B = 12
B+C = 18
A+C = 10

As A+B and A+C is less than B + C => Anne has the least amount of money.

Adding all three equations, A+B+B+C+A+C = 12+18+10 = 40
2A+2B+2C = 40
A+B+C = 20
Substituting B+C = 18
A+18 = 20
A = 2

Anne has $2.

Anne has the least amount of money and she has $2.

Find three consecutive numbers?

Consecutive numbers are whole numbers that follow in order such as 7, 8, 9, 10, 11, and 12. Find three consecutive numbers such that the sum of first and third is 118.

For three consecutive numbers the average of three will be the middle number given by sum of the numbers divided by 2.

i.e; 118/2 will be the middle number which is equal to 59.

The three consecutive numbers are 58, 59 and 60.

What number does DBC represent?

In the multiplication problem at the right, different letters stand for different digits, and ABC and DBC each represent a three-digit number. What number does DBC represent? (Two answers possible. Give one)

In the given problem C * C = _ C, the units digit is the same digit.
It can be possible only in 3 cases, C should be 1, 5 or 6.

(1) C = 1
If C = 1, ABC * 1 = ABC as its not ABC but DBC, C cannot be 1.

(2) C = 5
If C * C = 25
And B * 5 = B => B can only be 2 or 7.
Also A * C should be a single digit number D => A can be only be 1.

So ABC can be 125 or 175 => DBC can be 625 or 875.

DBC is 625 or 875.