Question 1 :
18 is taken away from 8 times of a number is 30. Find the number.
Question 2 :
The denominator of a fraction exceeds the numerator by 5. If 3 be added to both, the fraction becomes 3/4. Find the fraction.
Question 3 :
If thrice of A's age 6 years ago be subtracted from twice his present age, the result would be equal to his present age. Find A's present age.
Question 4 :
A number consists of two digits. The digit in the tens place is twice the digit in the units place. If 18 be subtracted from the number, the digits are reversed. Find the number.
Question 5 :
For a certain commodity, the demand equation giving demand "d" in kg, for a price "p" in dollars per kg. is d = 100(10 - p). The supply equation giving the supply "s" in kg. for a price "p" in dollars per kg is s = 75(p - 3). Find the equilibrium price.
Question 6 :
The fourth part of a number exceeds the sixth part by 4. Find the number.
Question 7 :
The width of the rectangle is 2/3 of its length. If the perimeter of the rectangle is 80 cm. Find its area.
Question 8 :
In a triangle, the second angle is 5° more than the first angle. And the third angle is three times of the first angle. Find the three angles of the triangle.
Question 9 :
If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes 1. In case, the numerator is decreased by 4 and the denominator by 2, it becomes 1/2. Find the fraction.
Question 10 :
A park charges $10 for adults and $5 for kids. How many many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of $3750 ?
Question 11 :
A number consists of three digits of which the middle one is zero and the sum of the other digits is 9. The number formed by interchanging the first and third digits is more than the original number by 297.Find the number.
Question 12 :
A manufacturer produces 80 units of a product at a cost of $22000 and 125 units at a cost of $28750. Assuming the cost curve to be linear, find the equation of the cost curve and then use it to estimate the cost of 95 units.
Question 13 :
Y is older than X by 7 years. 15 years back X's age was 3/4 of Y's age. Find the present their present ages.
Question 14 :
Difference between a number and its positive square root is 12. Find the number.
Question 15 :
A piece of iron rod costs $60. If the rod was 2 meter shorter and each meter costs $1 more, the cost would remain unchanged. What is the length of the rod ?
Question 16 :
Divide 25 in two parts so that sum of their reciprocals is 1/6.
Question 17 :
The hypotenuse of a right angled triangle is 20 cm. The difference between its other two sides is 4 cm. Find the length of the sides.
Question
18 :
The sides of an equilateral triangle are shortened by 12 units, 13 units and 14 units respectively and a right angle triangle is formed. Find the length of each side of the equilateral triangle.
Question 19 :
In a triangle, If the second angle is 20% more than the first angle and the third angle is 20% less than the first angle, then find the three angles of the triangle.
Question 20 :
This year, the chickens on a farm laid 30% less eggs than they did last year. If they had laid 3500 eggs this year, how many eggs did they lay last year ?
Answers
Question 1 :
18 is taken away from 8 times of
a number is 30. Find the number.
Answer :
Let 'x' be the number.
Given : 18 is taken away from 8 times of the number is 30
Then, we have
8x - 18 = 30
Add 18 to both sides.
8x = 48
Divide both sides by 8.
x = 6
So, the number is 6.
Question 2 :
The denominator of a fraction
exceeds the numerator by 5. If 3 be added to both, the fraction becomes 3/4. Find the fraction.
Answer :
Let 'x' be the numerator.
"The denominator of the fraction exceeds the numerator"
From the above information,
Fraction = x / (x + 5) ----------(1)
"If 3 be added to both, the fraction becomes 3 / 4"
From the above information, we have
(x+3) / (x + 5 + 3) = 3 / 4
Simplify.
(x + 3) / (x + 8) = 3/4
4(x + 3) = 3(x + 8)
4x + 12 = 3x + 24
x = 12
Plug x = 12 in (1)
Fraction = 12 / (12 + 5)
Fraction = 12 / 17
So, the required fraction is 12 / 17.
Question 3 :
If thrice of A's age 6 years ago be subtracted from twice his present age, the result would be equal to his present age. Find A's present age.
Answer :
Let 'x' be A's present age.
A's age 6 years ago = x - 6
Thrice of A's age 6 years ago = 3(x-6)
Twice his present age = 2x
Given : Thrice of A's age 6 years ago be subtracted from twice his present age, the result would be equal to his present age.
So, we have
2x - 3(x - 6) = x
Simplify.
2x - 3x + 18 = x
- x + 18 = x
18 = 2x
Divide both sides by 2.
9 = x
So, A's present age is 9 years.
Question 4 :
A number consists of two digits. The digit in the tens place is twice the digit in the units place. If 18 be subtracted from the number, the digits are reversed. Find the number.
Answer :
Let 'x' be the digit in units place.
Then, the digit in the tens place = 2x
So, the number is (2x)x.
Given : If 18 be subtracted from the number, the digits are reversed.
So, we have
(2x)x - 18 = x(2x)
(2x)x - 18 = x(2x)
10 ⋅ (2x) + 1 ⋅ x - 18 = 10 ⋅ x + 1 ⋅ (2x)
Simplify.
20x + x - 18 = 10x + 2x
21x - 18 = 12x
21x - 18 = 12x
9x = 18
Divide both sides by 9.
x = 2
The digit at the units place is 2.
Then, the digit at the tens place is
= 2 ⋅ 2
= 4
So, the required number is 42.
Question 5 :
For a certain commodity, the demand equation giving demand "d" in kg, for a price "p" in dollars per kg. is d = 100(10 -
p). The supply equation giving the supply "s" in kg. for a price "p" in dollars per kg is s = 75(p - 3). Find the equilibrium price.
Answer :
The equilibrium price is the market price where the quantity of goods demanded is equal to the quantity of goods supplied.
So, we have
d = s
100(10 - p) = 75(p - 3)
Simplify.
1000 - 100p = 75p - 225
1225 = 175p
Divide both sides by 175.
7 = p
So, the equilibrium price is $7.
Question 6 :
The fourth part of a number exceeds the sixth part by 4. Find the number.
Answer :
Let 'x' be the required number.
Fourth part of the number = x/4
Sixth part of the number = x/6
Given : The fourth part of a number exceeds the sixth part by 4.
x/4 - x/6 = 4
L.C.M of (4, 6) is 12.
(3x/12) - (2x/12) = 4
Simplify.
(3x - 2x) / 12 = 4
x / 12 = 4
Multiply both sides by 12.
x = 48
So, the required number is 48.
Question 7 :
The width of the rectangle is 2/3 of its length. If the perimeter of the rectangle is 80 cm. Find
its area.
Answer :
Let 'x' be the length of the rectangle.
Then, width of the rectangle is 2x / 3
Given : Perimeter is 80cm.
Perimeter = 80 cm
2 ⋅ (l + w) = 80
Divide both sides by 2.
l + w = 40
Plug l = x and w = 2x / 3.
x + 2x / 3 = 40
Simplify.
(3x + 2x) / 3 = 40
5x / 3 = 40
Multiply both sides by 3/5.
x = 24
The length is 24 cm.
Then, the width is
= 2x / 3
= (2 ⋅ 24) / 3
= 16 cm
Formula to find the area of a rectangle is
= l ⋅ w
Plug l = 24 and w = 16.
= 24 ⋅ 16
= 384
So, area of the rectangle is 384 square cm.
Question 8 :
In a triangle, the second angle is 5° more than the first angle. And the third angle is three times of the first angle. Find the three angles of the triangle.
Answer :
Let x° be the first angle.
Then, we have
the second angle = x° + 5°
third angle = 3 ⋅ x°
We know that the sum of three angle in any triangle is 180°.
x° + (x° + 5°) + (3 ⋅ x°) = 180°
x + x + 5 + 3x = 180
Simplify.
5x + 5 = 180
Subtract 5 from both sides.
5x = 175
Divide both sides by 5.
x = 35
The first angle is 35°.
The second angle is
= 35° + 5°
= 40°
The third angle is
= 3 ⋅ 45°
= 135°
So, the three angles of the triangle are 35°, 40° and 135°.
Question 9 :
If the numerator of a
fraction is increased by 2 and the denominator by 1, it becomes 1. In case, the numerator is decreased by 4 and the denominator by 2, it becomes 1/2. Find the fraction.
Answer :
Let 'x/y' be the fraction.
Given : If the numerator is increased by 2 and the denominator by 1, the fraction becomes 1.
So, we have
(x + 2) / (y + 1) = 1
Simplify.
(x + 2) / (y + 1) = 1
x + 2 = y + 1
x - y = - 1 ------(1)
Given : In case the numerator is decreased by 4 and the denominator by 2, the fraction becomes 1/2.
So, we have
(x - 4) / (y - 2) = 1 / 2
Simplify.
2(x - 4) = 1(y - 2)
2x - 8 = y - 2
2x - y = 6 ------(2)
Solving (1) and (2), we get
x = 7 and y = 8
So, the fraction is 7/8.
Question 10 :
A park charges $10 for adults and $5 for kids. How many many adults tickets and kids tickets were sold, if a total of 548 tickets were sold for a total of $3750 ?
Answer :
Let 'x' be the no. of adult tickets and "y' be the no. of kids tickets sold.
Given :A total of 548 tickets were sold
So, we have
x + y = 548 ------(1)
Given : Cost of each adult ticket is $10 and kid ticket is $5 and tickets were sold for a total of $3750.
So, we have
10x + 5y = 3750
2x + y = 750 ------(2)
Solving (1) & (2), we get
x = 202 and y = 346
So, the number of adults tickets sold is 202 and kids tickets is 346.
Question 11 :
A number consists of three digits of which the middle one is zero and the sum of the other digits is 9. The number formed by interchanging the first and third digits is more than the original number by 297.Find the number.
Answer :
Let 'x0y' be the three digit number. (As per the given information, middle digit is zero)
Given : The sum of the other digits is 9
x + y = 9 ------(1)
By interchanging the first and third digits, the number we get is
y0x
Given : The number formed by interchanging the first and third digits is more than the original number by 297
y0x - x0y = 297
(100y + x) - (100x + y) = 297
100y + x - 100x - y = 297
-99x + 99y = 297
- x + y = 3 ------(2)
Solving (1) & (2), we get
x = 3 and y = 6
So, we have
x0y = 306
So, the three digit number is 306.
Question 12 :
A manufacturer produces 80 units of a product at a cost of $22000 and 125 units at a cost of $28750. Assuming the cost curve to be linear, find the equation of the cost curve and then use it to estimate the cost of 95 units.
Answer :
Since the cost curve is linear, its equation will be
y = Ax + B.
(Here y = Total cost, x = no. of units)
Given : The total cost of 80 units of the product is $22000.
So, we have
22000 = 80A + B
80A + B = 22000 ------(1)
Given : The total cost of 125 units of the product is $28750.
So, we have
28750 = 125A + B
125A + B = 28750 ------(2)
Solving (1) and (2), we get
A =
150 and B = 10000
Then, the equation of the cost curve is
y = 150x + 10000 ------(3)
Estimate the cost of 95 units :
Plug x = 95 in (3).
(3)------> y = 150 ⋅ 95 + 10000
y = 14250 + 10000
y = 24250
So, the cost of 95 units is $24250.
Question 13 :
Y is older than X by 7 years. 15 years back X's
age was 3/4 of Y's age. Find the present their present ages.
Answer:
Let 'x' be the present age of X and 'y' be the present age of Y.
Given : Y is older than X by 7 years.
So, we have
y = x + 7 --------(1)
15 years back :
X's age = x - 15
Y's age = y - 15
Given : 15 years back X's age was 3/4 of Y's
age.
So, we have
(x - 15) = 3/4 ⋅ (y - 15)
Simplify.
4(x - 15) = 3(y - 15)
4x - 60 = 3y - 45
4x = 3y + 15 ------(2)
Solving (1) and (2), we get
x = 36 and y = 43
So, the present ages of "x" and "y" are 36 years and 43 years.
Question 14 :
Difference between a number and its positive square
root is 12. Find the number.
Answer :
Let 'x' be the required number.
Its positive square root is √x
Given : Difference between x and √x = 12
x - √x = 12
x - 12 = √x
(x - 12)2 = x
x2 - 24x + 144 = x
x2 - 25x + 144 = 0
(x - 9)(x - 16) = 0
x = 9 or x = 16
x = 9 does not satisfy the condition given in the question.
So, the required number is 16.
Question 15 :
A piece of iron rod costs $60. If the rod was 2 meter shorter and each meter costs $1 more, the cost would remain unchanged. What is the length of the rod?
Answer :
Let 'x' be the length of the given rod.
Then the length of the rod 2 meter shorter is (x - 2) and the total cost of both the rods is $60 (Because cost would remain unchanged).
Cost of one meter of the given rod is
= 60 / x
Cost of one meter of the rod which is 2 meter shorter is
= 60 / (x - 2)
Given : If the rod was 2 meter shorter and each meter costs $1 more.
That is, 60/(x-2) is $1 more than 60/x.
[60 / (x - 2)] - [60 / x] = 1
Simplify.
[60x - 60(x - 2)] / [x(x - 2)] = 1
[60x - 60x + 120] / [x2 - 2x] = 1
120 / (x2 - 2x) = 1
120 = x2 - 2x
0 = x2 + 2x - 120
x2 + 2x - 120 = 0
(x
+ 10)(x - 12) = 0
x = - 10 or x = 12
Because length can not be a negative number, we can ignore "- 10".
So, the length of the given rod is 12 m.
Question 16 :
Divide 25 in two parts so that sum of their reciprocals is 1/6.
Answer :
Let 'x' be one of the parts of 25. Then the other part is (25 - x).
Given : Sum of the reciprocals of the parts is 1/6.
Then, we have
1/x + 1/(25 - x) = 1/6
Simplify.
(25 - x + x) / x(25 - x) = 1/6
25 / (25x - x2) = 1/6
6(25) = 25x - x2
150 = 25x - x2
x2 - 25x + 150 = 0
(x - 15)(x - 10) = 0
x = 15 or x = 10
When x = 15,
25 - x = 25 - 15
25 - x = 10
When x = 10,
25 - x = 25 - 10
25 - x = 15
So, the two parts of
the 25 are 10 and 15.
Question 17 :
The hypotenuse of a right angled triangle is 20 cm. The difference between its other two sides is 4 cm. Find the length of the sides.
Answer :
Let 'x' and "x + 4" be the lengths of other two sides.
Using Pythagorean theorem, we have
(x + 4)x2 + x2 = 20x2
Simplify.
x2 + 8x + 16 + x2 = 400
2x2 + 8x + 16 = 400
Subtract 400 from both sides.
2x2 + 8x - 384 = 0
Divide both sides by 2.
x2 + 4x - 192 = 0
(x + 16)(x - 12) = 0
x = -16 or x = 12
x = -16 can not be accepted. Because length can not be negative.
If x = 12,
x + 4 = 12 + 4 = 16
So, the other two sides of the triangle are 12 cm and 16 cm.
Question 18 :
The sides of an equilateral triangle are shortened by 12 units, 13 units and 14 units respectively and a right angle triangle is formed. Find the length of each side of the equilateral triangle.
Answer :
Let 'x' be the length of each side of the equilateral triangle.
Then, the sides of the right angle triangle are
(x - 12), (x - 13) and (x - 14)
In the above three sides, the side represented by (x - 12) is hypotenuse (Because that is the longest side).
Using Pythagorean theorem, we have
(x - 12)2 = (x - 13)2 + (x - 14)2
x2 - 24x + 144 = x2 - 26x + 169 + x2 - 28x + 196
x2 - 30x + 221 = 0
(x - 13)(x - 17) = 0
x = 13 or x = 17.
x = 13 can not be accepted.
Because, if x = 13, the side represented by (x - 14) will be negative.
So, the side of the equilateral triangle is 17 units.
Question 19 :
In a triangle, If the second angle is 20% more than the first angle and the third angle is 20% less than the first angle, then find the three angles of the triangle.
Answer :
Let 'x' be the first angle.
Then, the second angle = 120% of x = 1.2x
The third angle = 80% of x = 0.8x
We know that,
the sum of the three angles of a triangle = 180°
x + 1.2x + 0.8x = 180°
3x = 180°
x = 60°
The first angle = 60°
The second angle = 1.2(60) = 72°
The third angle = 0.8(60) = 48°
So, the three angles of a triangle are 60°, 72° and 48°.
Question 20 :
This year, the chickens on a farm laid 30% less eggs than they did last year. If they had laid 3500 eggs this year, how many eggs did they lay last year ?
Answer :
Let 'x' be the number of eggs laid last year.
Given : This year, the chickens laid 30% less eggs than they did last year and they laid 3500 eggs this year.
Then, we have
(100 - 30)% of x = 3500
70% of x = 3500
0.7 ⋅ x = 3500
Divide each side by 0.7
x = 5000
So, the chickens laid 5000 eggs last year.
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