Problems on Money -
Example.1) The monthly income of A & B are in the ratio 2 : 5, and their monthly expenditures are in the ratio 1 : 3. If A saves $ 4000 and B saves $ 3000 per month, find the monthly income of each.
Ans.) As per the given condition, the monthly income of A & B are in the ratio 2 : 5, their monthly expenditures are in the ratio 1 : 3.
So, let the monthly income of A & B be $ 2x and $ 5x respectively and let their monthly expenditures be $ y and $ 3y respectively. Then,
Monthly saving of A = $ (2x – y)
Monthly savings of B = $ (5x – 3y)
As per the given condition –
(2x – y) = 4000 ………………(i)
(5x – 3y) = 3000 ………………(ii)
Multiply (i) by 5, and we get –
10x – 5y = 20000 ………………(iii)
Multiply (ii) by 2, and we get –
10x – 6y = 6000 ……………..(iv)
Now, we will subtract (iii) from (iv), and we get –
10x – 6y = 6000
10x – 5y = 20000
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- y = - 14000
Or, y = 14000
Now, we will substitute y in the equation (i)
(2x – y) = 4000
=> 2x – 14000 = 4000
=> 2x = 18000
=> x = 9000
So, A’s income => 2x = 2 X 9000 = $ 18000
And, B’s income => 5x = 5 X 9000 = $ 45000 (Ans.)
Example.2) Taxi charges in a city consist of fixed charges and the remaining depending upon the distance traveled in kilometers. If a person travels 70 km, he pays $ 1130 and for traveling 100 km, he pays $ 1550. Find the fixed charges and the rate per km.
Ans.) Let the fixed charges be $ x and the other charges by $ y per km.
Then, x + 70y = 1130 ………………(i)
And, x + 100y = 1550 ……………….(ii)
Now, we will subtract (i) from (ii), we get -
x + 100y = 1550
x + 70y = 1130
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30y = 420
=> y = 14
Now, we will substitute the value of ‘y’ in (i), and we find –
x + 70y = 1130
=> x + (70 X 14) = 1130
=> x = 1130 – 980 = 150
So, fixed charges = $ 150 and rate = $ 14 per km (Ans.)
Example.3) A part of monthly hostel charges in a school is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 23 days, he has to pay $ 4730 as hostel charges, where as a student B, who takes food for 30 days, pays $ 5850 as hostel charges. Find the fixed charges and the cost of food per day.
Ans.) Let the fixed charges per month be $ x and cost of food per day be $ y. Then,
x + 23y = 4730 ……………..(i)
and, x + 30y = 5850 ……………….(ii)
On subtracting (i) from (ii), we get –
x + 30y = 5850
x + 23y = 4730
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7y = 1120
=> y = 160
Substituting y = 160 in (i), and we get –
x + 23y = 4730
=> x + (23 X 160) = 4730
=> x = 4730 – 3680 = 1050
So, fixed charges per month = $ 1050, and the cost of food per day (Ans.)
Example.4) Each one of A and B, has some money. If A gives $ 60 to B, then B will have twice the money left with A. but if B gives $ 30 to A, then A will have thrice as much as is left with B. How much money does each have?
Ans.) Let A and B have $ x and $ y respectively.
Case.1) When A gives $ 60 to B : -
Then money with A = $ (x – 60)
And, money with B = $ (y + 60)
So, (y + 60) = 2(x – 60)
=> 2x – y = 180 …………….(i)
Case.2) When B gives $ 30 to A :-
Then money with A = $ (x + 30)
And, money with B = $ (y – 30)
So, (x + 30) = 3(y – 30)
=> x + 30 = 3y – 90
=> x – 3y = - 120 ………………(ii)
Multiplying (ii) by 2, and we get –
2x – 6y = - 240 ………………(iii)
Now we will subtract (iii) from (i), we
2x – y = 180
2x – 6y = - 240
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5 y = 420
=> y = 84
Now, we will substitute the value of ‘y’ in (i), and we will get –
2x – y = 180
=> 2x – 84 = 180
=> 2x = 180 + 84 = 264
=> x = 132
Hence, A has $ 132 money and B has $ 84 money. (Ans.)
Example.5) 18 pen and 32 pencils together cost $ 642, while 32 pens and 18 pencils together cost $ 908. Find the cost of each pen and that of each pencil.
Ans.) Let the cost of each pen be $ x and that of each pencil be $ y.
As per the given condition –
18x + 32y = 642 ………………..(i)
32x + 18y = 908 …………………(ii)
Adding (i) and (ii), and we get –
18x + 32y = 642
32x + 18y = 908
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50x + 50y = 1550
=> 50 (x + y) = 1550
=> x + y = 31 ……………………..(iii)
Now, we will subtract (i) from (ii), and we find –
32x + 18y = 908
18x + 32y = 642
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14x – 14y = 266
=> 14 (x – y) = 266
=> x – y = 19 ……………………(iv)
Now, we will add (iii) & (iv), then we will find –
x + y = 31
x – y = 19
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2x = 50
=> x = 25
Subtracting the value of x in (iii), and we get –
x + y = 31
=> 25 + y = 31
=> y = 31 – 25 = 6
So, the cost of each pen is $ 25 and cost of each pencil $ 6 (Ans.)