What is the difference between the second number and the first number?

 The sum of two numbers is 76. One-third of the first number plus half of the second number is 34. What is the difference between the second number and the first number?

Let us define the first number as x and the second number as y. We can write the following two equations

x + y = 76 .......... (a). This equation is derived from the statement "the sum of the 2 numbers is 76"

1/3 x + 1/2 y = 34 ............ (b). This equation is derived from the statement " 1/3 of the first number plus 1/2 the second number is 34. 

Now let's multiply (b) by 6

2x + 3y = 204 ............ (c)

Now multiply (a) by 2

2x + 2y = 152 ............ (d)

Subtract d from c to find out what the second number (y) is. 

y = 204 - 152 = 52

Now plug y into equation (a) to solve for the first number (x). 

x + 52 = 76

Subtracting 52 from both sides we get:

x = 76-52 = 24

Now we know that the first number is 24 and the second number is 52, but we are asked for the difference between the two numbers. 

52 - 24 = 28

Ans: 28

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.

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 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.

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 coins of each kind were in the change?

A dollar was changed into 16 coins consisting of just nickels and dimes. How many coins of each kind were in the change?

Total amount of money = 1$ = 100cents.
Total number of coins = 16
Available denominations of coins = Nickels(N:5) and Dimes(D:10)

Given, N + D = 16
and 5N + 10D = 100

Solving the two equations:

10N+10D=160
5N+10D=100
5N = 60
N=12

If N =12,
12+D = 16
D = 16-12 = 4

There were 12 nickels and 4 dimes in the change.

How many plums have the weight of one pear?

Thirteen plums weigh as much as two apples and one pear. Four plums and one apple have the same weight as one pear. How many plums have the weight of one pear?

Let L represents Plums, A represents apples and P represents pears.

13L = 2A+1P
4L+1A=1P
1A=1P-4L

13L=2(P-4L)+1P
13L = 2P - 8L + 1P
13L+8L = 3P
21L=3P
7L=P

7 plums have the weight of one pear.

How many tests did he fail?

During a school year, a student was given an award of 25¢ for each math test he passed and was fined 50¢ for each math test he failed. At the end of the school year, the student had passed 7 times as many tests as he had failed, and received $3.75. How many tests did he fail?

Let P be the number of tests the student passed and F be the number of tests the student failed. 
As the student has passed 7 times as many tests as he had failed,
P = 7F ------(1)

As he has received a total of $3.75 = 375¢
25P - 50F = 375 -------(2)

Substituting for P in eqn(2)
25 * 7F - 50F = 375
175F - 50F = 375
125F = 375
F = 3

The student failed in 3 tests.

What is the sum of the two numbers?

The product of two numbers is 144 and their difference is 10. What is the sum of the two numbers?

MetroEye: Refer KeyFormulas

As per formula, \( (A+B)^2 = (A-B)^2 + 4AB\)

=> \( (A+B)^2 = 10^2 + 4\cdot 144 \) = 100+576 = 676
=> A+B = \(\sqrt{676}\) = 26

The sum of two numbers is 26.

What is the number?

If 20 is added to one-third of a number, the result is the double of the number. What is the number?

Let N be the number. 

So 20+(1/3)N=2N

Rearranging like terms;

2N-(1/3)N = 20
ie (5/3)N=20
=>N= 20*3/5 = 12

The number is 12.

How many boxes of each size did he use?

A dealer packages marbles in two different box sizes. One size holds 5 marbles and the other size holds 12 marbles. If the dealer packaged 99 marbles and used more than 10 boxes, how many boxes of each size did he use?

Let A represent number of boxes holding 5 marbles and B represent number of boxes holding 12 marbles.

=> 5A+12B = 99
& A+B > 10

A = \(\dfrac{99-12B}{5}\)

For A to be a whole number 99-12B should be divisible by 5. For any number to be divisible by 5, the units digit should be 5 or 0. 

i.e; 12B should have 4 or 9 in units place as 9-4=5 or 9-9=0
As no multiple of 12 will have 9 in units place, 12B should have 4 in units place.
=> B can be 2 as 12B=24 or B can be 7 as 12B=84

If B=2, A = \(\dfrac{99-12\cdot 2}{5}\) = \(\dfrac{75}{5}\)=15

If B=7, A = \(\dfrac{99-12\cdot 7}{5}\) = \(\dfrac{15}{5}\)=3

As A+B > 10 => A=15, B=2

The dealer used 15 boxes of 5 marbles size and 2 boxes of 12 marbles size.

Divisible by 9

The four digit numeral 3AA1 is divisible by 9.What digit does A represent?

MetroEye: If a number is divisible by 9, the sum of digits should also be divisible by 9

=> 3+A+A+1=4+2A should be divisible by 9
If 4+2A=9, then A=2.5 which is not integer.
If 4+2A=18, then A=7 which is an integer.

So A represents digit 7.

Average speed for the entire trip?

A motorist made a 60 mile trip averaging 20 miles per hour. On the return trip, he averaged 30 miles per hour. What is the motorist's average speed for the entire trip?

Average speed = Total distance travelled / Total time taken

Total distance for the trip = 60+60=120 miles
Total time taken for the trip = 60/20+60/30=3+2=5 hours.

So the average speed = 120/5=24 mph.

Motorist's average speed for the entire trip is 24 miles per hour.

What is the 100th number in the sequence?

If we count by 3s starting with 1, the following sequence is obtained: 1,4,7,10, ... What is the 100th number in the sequence?

MetroEye: t_n=a+(n-1)d
where t_n = n^th term of the sequence
a = first term of the sequence
n = number of terms in the sequence
d = common difference of the sequence

In the given sequence, a=1,n=100,d=3 
=> t₁₀₀=1+(100-1)3=298
So the 100^th term of the sequence is 298

Greatest number of rectangular cards that can be cut?

Rectangular cards, 2 inches by 3 inches, are cut from a rectangular sheet 2 feet by 3 feet. What is the greatest number of cards that can be cut from the sheet?

Area of the rectangular sheet = 2 feet X 3 feet = 2×12×3×12=864 sq.inch
Area of the rectangular card = 2 inch X 3 inch = 6 sq.inch

Number of cards that can be cut = 864÷6 = 144

So the greatest number of cards that can be cut from the sheet is 144.

Average score for all four bowling games?

In three bowling games, Alice scores 139, 143 and 144. What score will Alice need in a fourth game in order to have an average score of 145 for all four games?

MetroEye: Average Score = Sum of scores / Number of games

i.e; 145 = (139+143+144+S)/4
426+S = 145×4
S = 154

Hence Alice will need a score of 154 in the fourth game.

How long it would take to complete the work at the same rate?

A work crew of 3 people requires 3 weeks and 2 days to do a certain job. How long would it take a work crew of 4 people to do the same job if each person of both crews works at the same rate as each of the others? Note : each week contains six work days.

MetroEye: This problem is  the work-person-rate problem.
i.e; W₁/(M₁×H₁) = W₂/(M₂×H₂)
where W is work to be done,  M is the number of persons required, Hour is time required. 

Here as the work done is same W₁ = W₂
=> M₁×H_{1 =} M₂×H₂
H₁= 3 weeks 2 days = 3×6+2=20 days
=> 3×20= 4 × H₂
H₂ = 60÷4 = 15 days = 2 weeks 3 days
So 4 people should work 2 weeks and 3 days to do the same job.

How much does one pencil cost?

In a stationery store, pencils have one price and pens have another price. Two pencils and three pens cost 78¢. But three pencils and two pens cost 72¢. How much does one pencil cost?

Let cost of one pencil = P
and cost of one pen = N

Two pencils and three pens cost 78¢
i.e; 2P+3N=78¢ .........(1)
Three pencils and two pens cost 72¢
i.e; 3P+2N=72¢ .........(2)

Solving equations (1) and (2)
(1) × 2 is
4P+6N=156 ......(3)
(2) × 3 is
9P+6N=216 ......(4)

(4) - (3) 
5P=60
=> P=12

Cost of one pencil is 12¢.

How many even numbers are there between?

How many even numbers are there between 66 and 71 ?

MetroEye: 'between' means excluding start and end numbers. Number of integers between  N1 and N2  will be  N2-N1-1

If the start number is odd and end number is even (vice versa) then the count of in between numbers will be even.

If the start and end numbers are even (or odd) then the count of in between numbers will be odd, giving the 'other' one more than the start and end type

Eg : 66  & 70  , There are 3 numbers in between ; 1 even and 2 odd.
        67 & 71  , there are  3 numbers in between ; 1 odd and 2 even.

Here 71-66-1 = 4.

So four numbers in between. 2 of them odd and 2 of them even.

There are 2 even numbers between 66 and 71.

Michelle's Number Recycling Machine

Michelle's Number Recycling Machine obeys exactly two rules:

1. If an inserted number has exactly 1 digit, double the number.
2. If an inserted number has exactly 2 digits, compute the sum of the digits.

The first number Michelle inserts is 1. Then every answer she gets is inserted back into the machine until fifty numbers are inserted. What is the fiftieth number to be inserted?

As per the image, after 9 steps input again becomes 1 and the same steps through 1 to 9 repeats.
To get 50th input, steps 1 to 9 has to be repeated 5 times and then steps 1 to 5 once.
i.e; After 45 th step (9 steps * 5 times) input will be again 1.
Then we need 5 more steps to reach 50th input which is the output of step 5 = 7

Fiftieth number to be inserted will be 7.