Given A+B+C = AB
AB means 10th digit is A and B is the units digits, AB is 10A+Bi.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.
Showing posts with label Numbers. Show all posts
Showing posts with label Numbers. Show all posts
Sunday
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.
Given A+B+C = AB
AB means 10th digit is A and B is the units digits, AB is 10A+Bi.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.
Tuesday
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.
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.
Monday
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:
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.
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.
Saturday
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.
Friday
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?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 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.
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.
Thursday
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.
Wednesday
What is the smallest number that has above properties?
A number has a remainder of 1 when divided by 4, a remainder of 2 when divided by 5 and a remainder of 3 when divided by 6. What is the smallest number that has above properties?
Let N be the number.
(1) N/4 gives a remainder of 1 => N+3 divisible by 4
(2) N/5 gives a remainder of 2 => N+3 divisible by 5
(3) N/6 gives a remainder of 3 => N+3 divisible by 6
From the above conclusions, N+3 should be divisible by 4, 5 and 6.
The lowest common multiple of 4, 5 and 6 is 60.
i.e; N+3 = 60
N=57.
The smallest number with the mentioned properties is 57.
Let N be the number.
(1) N/4 gives a remainder of 1 => N+3 divisible by 4
(2) N/5 gives a remainder of 2 => N+3 divisible by 5
(3) N/6 gives a remainder of 3 => N+3 divisible by 6
From the above conclusions, N+3 should be divisible by 4, 5 and 6.
The lowest common multiple of 4, 5 and 6 is 60.
i.e; N+3 = 60
N=57.
The smallest number with the mentioned properties is 57.
Tuesday
Find the product?
In the multiplication problem at the right, each blank space represents a missing digit.
Find the product?
To have the first product i.e; 4 _ _ * 7 = _ _ 82
=> Units place should be 2, the multiple of 7 with units place 2 is 42 which is 6*7
i.e; 4 _ 6*7 = _ _ 82
10th place is 8 and we have a carry over of 4 from units place
=> _ *7 = _ 4
multiple of 7 with 4 in the units place is 14 which is 2*7
i.e; 426*7 = 2982
Second product is 426*_ = 12 _ _
=> Thousands and hundreds place is 12, multiple of 4 which is 12 or less. i.e; 4*3 or less.
i.e; the number should be 3 or less.
Substituting 3
426*3=1278
So the total product is 426*37 = 15762
The product is 15762
To have the first product i.e; 4 _ _ * 7 = _ _ 82
=> Units place should be 2, the multiple of 7 with units place 2 is 42 which is 6*7
i.e; 4 _ 6*7 = _ _ 82
10th place is 8 and we have a carry over of 4 from units place
=> _ *7 = _ 4
multiple of 7 with 4 in the units place is 14 which is 2*7
i.e; 426*7 = 2982
Second product is 426*_ = 12 _ _
=> Thousands and hundreds place is 12, multiple of 4 which is 12 or less. i.e; 4*3 or less.
i.e; the number should be 3 or less.
Substituting 3
426*3=1278
So the total product is 426*37 = 15762
The product is 15762
Thursday
What digits do the letters H,E and A represent?
In the problem, A represents a single digit i.e; A < 9.
=> 4H<10 =>H<3
i.e; H should be 1 or 2.
In units column, 4E should have H in units place in the result. i.e; 1 or 2
As 4E is always an even number, H cannot be 1. => H = 2
If H = 2, E can be 3 or 8 for 4E to end in 2.
If E = 8, the result will be a three-digit number => E = 3
So if H = 2 and E = 3
A = 9
The letters H = 2, E = 3, A = 9.
Wednesday
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?
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.
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.
Tuesday
In the magic-square at the right, what value should X have?
In the magic-square at the right, five more numbers can be placed in the boxes so that the sum of the three numbers in each row, in each column, and in each diagonal is always the same. What value should X have?
From the first column, the sum = 15+50+25 = 90
From the first row sum = 15+A+35 = 90
=>A = 90-35-15 = 40
From the right diagonal sum = 35+B+25=90
=>B = 90-35-25 = 30
From middle column, sum = A+B+X = 90
=> X = 90-A-B = 90-40-30 = 20
The value of X is 20.
15
|
35
| |
50
| ||
25
|
X
|
From the first column, the sum = 15+50+25 = 90
15
|
A
|
35
|
50
|
B
| |
25
|
X
|
From the first row sum = 15+A+35 = 90
=>A = 90-35-15 = 40
From the right diagonal sum = 35+B+25=90
=>B = 90-35-25 = 30
From middle column, sum = A+B+X = 90
=> X = 90-A-B = 90-40-30 = 20
The value of X is 20.
Monday
How many terminal zeroes will N have?
If a number ends in zero, the zeros are called terminal zeros. For example, 520,000 has four terminal zeroes, but 502,000 has just three terminal zeros. Let N equal the product of all counting numbers from 1 through 20.
N = 1×2×3×4× ... ×20
How many terminal zeroes will N have when it is written in standard form?
Terminal zeroes are obtained when multiplied by 10. So from numbers 1 to 20, we have two tens i.e; 10 and 20 => 2 zeroes
We have two 5s and two 2s i.e; 5,15 and 2,12 when multiplied will give two more zeroes.
N will have a total of 4 terminal zeros in the product.
N = 1×2×3×4× ... ×20
How many terminal zeroes will N have when it is written in standard form?
Terminal zeroes are obtained when multiplied by 10. So from numbers 1 to 20, we have two tens i.e; 10 and 20 => 2 zeroes
We have two 5s and two 2s i.e; 5,15 and 2,12 when multiplied will give two more zeroes.
N will have a total of 4 terminal zeros in the product.
Saturday
BasicMath01: Numbers
Numbers
Numbers are basically a set of standard symbols that represents quantities. These symbols are known as digits.
Natural Numbers
The natural numbers are counting numbers. 1,2,3,4……
Whole Numbers
When zero is included into natural numbers, the set becomes whole numbers.
0,1,2,3,4……
Integers
The whole numbers and their negatives makes up the set of integers.
Rational Numbers
Numbers that can be written as ratio of two integers. i.e; Fractions.
Irrational Numbers
Numbers that cannot be written as ratio of two integers.
Real Numbers
All the numbers that can be placed on a number line,which is a group of numbers arranged in a straight line.
All these sets are represented in Fig1:
Positive Numbers
A number that is more than zero. These are the numbers on the right hand side of the number line.
Negative Numbers
A number that is less than zero. These are the numbers on the left hand side of the number line.
Even Numbers
Any integer which is divisible by 2.
Odd Numbers
Any integer which when divided by 2 gives remainder 1.
MetroStation
Zero is even number.
Zero is neither positive nor negative.
Fractions are neither even nor odd.
Integer Arithmetic
Addition(+)
Numbers are added together to find their total. The result is called the SUM.
Subtraction(-)
A number is subtracted from another number to find what is left.This is known as DIFFERENCE.
Multiplication(×)
A number is added to itself a number of times. The result is called the PRODUCT.
Division(÷)
How many times one number fits into another number.
Dividend : The number that is being divided or shared by another number.
Divisor : The number that is being used to divide.
Quotient : The result of the division.
Remainder : The number that is left over.
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