Exercise Problem and Solution Part: 5 (Equation, Solving Equations, Systems of liner equations, Quadratic equations, Inequalities)

Question: 31 The sum of square of two numbers is 80 and the square of their difference is 16. Determine the product of the two numbers.

Explore: Let, the numbers be x & y.
According to the question,
X ² + Y ² = 80 and
(X+Y) ²= 16
=> X ²+ Y ²- 2XY = 16
=> 80 - 2XY = 16
=> 2xy = 80 -16 =64
=> xy = 64/2 = 32
Answer: Product of the two numbers is 32

Question: 32 A number consists of two digits, The sum of the digits is 15. If 27 is subtracted from the number, its digits are interchanged. Find the number ?
Explore: Let, the unit's digit be x.
:. Then ten's digit is (15 - X)
the number is 10(15 - X) + X
= 150 - 10X + X
= 150 - 9X
According to the question,
     150 - 9X - 27 = 10X + (15 - X)
=> 123 - 9X = 9X + 15
=> 9X +9X = 123 - 15 = 105
=> 18X = 108
=> X = 108/18 = 6
:. The number is (150 - 9 x 6)
= 150 - 54
= 96
Answer : The number is 69

Question: 33 The temperature in degrees Celsius (C) can be converted to temperature in degrees Fahrenheit (F) by the formula F = (9/5)C + 32. What is the temperature at which F = C ?
Explore:
     F = (9/5)C + 32
=> C = (9/5)C + 32 [F = C]
=> C - (9/5)C = 32
=> (5C - 9C)/5 = 32
=> (- 4C)/5 = 32
=> C = (32 x 5)/(- 4) = - 40 ⁰
Answer : - 40 ⁰

Question: 34 Sum of two numbers is 5 and their product is 4. The what is the differene between the numbers?
Option: (a) 2 (b) 4 (c) 5 (d) 3 (e) None
Explore: Let, X & Y be the two numbers. So, according to the problem
X + Y = 5 ......................................(1)
XY = 4 ..........................................(2)
From (1),
X + Y = 5 => X = 5 - Y
:. (5 -Y).Y = 4 [Putting the value of X from (1) in (2)]
Or, 5y - y²= 4
Or, y² - 5y + 4 = 0
Or, y²- 4y -y + 4 =0  [Using middle term technique]
Or, y(y - 4) - 1(y-4) = 0
Or, (y - 4)(y - 1) = 0
=> y = 1 or 4
and X = 5 - 1 = 4 Or, X = 5 - 4 = 1
We have to find out X - Y or Y - X
:. X - Y or Y - X = 3
Answer : (d)

Question: 35 Mr. Quddus rents a room for n months. He pays $ K for each of the first 7 months, and half that rate for each additional month. Find the total change if n > 7.
Option: (a) 7k + 2kn (b) (7k+n)/2 (c) (k+27)/(n-7) (d) (k+k(n-7))/2 (e) 7k(k(n-7))/2
Explore: Rent for the 1th K month 7k
We have (n-7) months and for each of these months, the rate is k/2. so, here the rate is k/2(n-7)
:. Total rent = 7k + (k(n-7))/2
Answer: (e)

Question: 36 In a picnic there were 240 persons. There were 20 more men then women and there were 20 more adults than children. How many men were there in the picnic?
 Option: (a) 240 (b) 75 (c) 110 (d) 130 (e) 200
Explore: Let, m, w & c be the number of men, women & children, respectively. According to the problem,
m = w + 20
(m + w) = c + 20
& m + w + c = 240
:. w + 20 + w + c + 240
Or, 2w + (c + 20) = 240
Or, 2w + m + w = 240
Or, 3w + m = 240
Or, 3w + w + 20 = 240
Or, 4w = 240 - 20 = 220
Or, w = 220/4 = 55
:. m = 75
Answer : (b)

Question: 37 3 apples and 4 oranges cost $ 32. 4 apples and 3 oranges cost $ 31. 1 apple, 1 orange and 1 papaya cost $ 28. What is the cost of papaya in $ ?
Option: (a) 19 (b) 18 (c) 20 (d) 18 (e) 17
Explore: We can write three equations from the given information
3a + 4b = 32 ........................... (1)
4a + 3b = 31 ........................... (2)
a + b + c = 28 ......................... (3)
(1) + (2) => 7(a +b) = 63 Or, a + b =9
:. a + b + c = 28 => 9 + c = 28
=> c = 19
Answer : (a)

Question: 38 X + Y = 1, Y + Z = - 1, Z + X = 2. What is the value of X - Y - Z ?
Option: (a) 1 (b) 0 (c) -1 (d) 3 (e) None
Explore: The ordinary way of solving this problem would be to find the individual values of x, y, z & then calculate (x - y - z). But here too we have an easier way.
2X +2Y + 2Z = 1- 1 + 2 = 2
:. X + Y + Z = 1 ....................(1)
Given Y + Z = - 1
:. 2 Y + 2 Z = - 2 ....................(2)
:. (1) - (2) gives us
X - Y - Z = 1 - (- 2) = 1 + 2 = 3
Answer : (d)

Question: 39 In a certain brick wall, each row of bricks above the bottom row contains one less brick than the row just below it. If there are 5 rows in all and a total to 75 bricks in the wall, how many bricks does the bottom wall contain?
Explore: Let, the bottom wall contain X bricks.
According to the question,
     X + (X - 1) + (X - 2) + (X - 3) +(X - 4) = 75
=> 5X - 10 = 75
=> 5X = 75 + 10
=> X = 85/5
=> X = 17
Answer : the bottom row contains 17 bricks.

Question: 40 A manager has budget of $ 60,000 for giving salary increment to 4 full time and 2 part time employees. Each of the full time employees receives the same increment, which is twice the increment that each of the part time employees received. What is the amount of the increment that each of the full time employees received?
Explore: Let, the full time employee received increment $ X. According to the question,
     4X + 2 x X/2 = 60,000
=> 4X + X = 60000
=> 5X = 60000
=> X = 60000/5
=> X = 12000
Answer : The amount of the increment is $ 12000

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