HIRE PURCHASE AND
INSTALLMENT BUYING
EXERCISE 2.3.5
1.A wash machine costs 10,200 cash down. It was bought by paying a down
payment of 2,000 and the balance was agreed to be paid in 6 equal monthly installments of 1,500 each find the rate of interest.
Solution:
Cost price
Down payment
Balance
= 10,200.00
= 2,000.00
= 10,200.002,000.00
8,200.00
Number of installments = 6
Amount of each installment (I)
= 1,500.00
Amount paid in 6 installments (n)
= 1,500.x6
9,000.00
Excess amount paid = 9,000 – 8,200
E = 800.00
Rate of interest =
2400E
n[(n + 1 )I – 2E]
2400x800
= 6[(6 + 1) 1500 – 2x800]
=
2400x800
6(7x1500 – 1600)
=
2400x800
6(10500 – 1600)
=
2400x800
6x8900
=35.95%
2. The cost of an android mobile phone is 8,990. Joseph bought it by paying
500Cash down and the balance he agreed to pay in 10 monthly installments
of 900each.Nizam bought the same phone by initially paying 900 and the
Remaining balance is 8 installments of 1,200 each.
Who has paid more rate of interest?
Solution:
Cost price of the phone = 8,990.00
(i) cash down payment
Made by Joseph = 500
Balance
= 8,990.00
500.00
8,490.00
Number of installments (n) =10
Amount of each installments (i) = 900.00
Amount paid in installments = 900x10
= 9,000.00
Extra amount paid =
9,000.00 –
8,490.00
510.00
2400E
Rate of interest
=
N[(n+1)I – 2E]
=
2400x510
10(11x900- 2x510)
2400x510
10[(10+1)900 – 2x510]
=
2,400x510
10x8880
= 13.78%
(ii) Cash down payment
made by Nizam =
Balance
=
900.00
8,990.00
900.00
8090.00
Number of installments(n) = 8
Amount of each installments(i) = 1,200.00
Amount paid in installments =
1,200x8
=9,600.00
Extra amount paid(E) = 9,600.00
8,090.00
1,510.00
2,400E
Rate of interest =
N[(n+1)I – 2E]
=
=
2400x1510
8(10,800 – 3,020)
2400x1510
8x7780
= 58.226% = 58.23%
Nizam is paying a higher rate of interest.
3. The cost of a motor bike is 48,000.The company offers it in 30 months of
Equal Installments at 10% rate of interest. Find the equated monthly
installment.
Solution:
R = 48,000.00
R = 10%
N= 30(number of installments)
P(2nR+2400)
Monthly installment I = N[2,400+(n+1)R]
48,000(2x30x10+2,400)
=
30[2,400+(30-1)10]
48,000(600+2,400)
= 30(2,400+290)
= 48,000x3,000 = 48,00,000
30x2,690
2,690
=
4,80,000
269
= 1784.38
4. The cost of a set of home appliances is 36,000. Siri wants to buy them
Under a scheme of 0% interest and by paying 3EMI in advance. The firm
Charges 3% as processing charges. Find the EMI and the installment for a
Period of 24 months.
Solution:
Cost of the set of home appliances (P)
=
36,000.00
Number of installments (n) = 24
P
Amount of each installment = n
= 36,000
24
=
1,500.00
Amount paid in advance = SEMI = 1500x3
= 4,500.00
Processing charge at the rate of 3% =
36,000x3
100
=
1,080.00
The total amount paid
= 1,500x24+1,080
= 36,000+1,080
=
37080
RATIO AND PROPORTION
EXERCISE 2.4.1
1.Write each of these ratios in the simplest form.
(i) 2:6 (II)24:4 (iii) 14:21 (IV) 20: 100
(v) 18:24
(Vi) 22:77
Solution:
(i)
2:6
= 1:3
(ii)
(iii)
(iv)
(v)
(vi)
24:4 = 6:1
14:21 = 2:3
20:100 = 1:5
18:24 = 3:4
22:77 = 2:7
(dividing both by 2)
(dividing both by 2)
(dividing both by 7)
(dividing both by 2)
(dividing both by 6)
(dividing both by 11)
2. A shop-keeper mixes 600ml of orange juice with 900ml of apple juice to make
A fruit drink. Write the ratio of orange juice to apple juice in the fruit drink in
Its simplest form.
Solution:
Ratio of volumes of
Orange juice and apple juice O:A
= 600:900
= 6:9
= 2:3
3. a builder mixes 10 shovels of cement with 25 shovels of sand. Write the ratio
Of cement to sand.
Solution:
Ratio of cement to sand = 10 shovels :25 shovels
4.In a school there are 850 pupils and 40 teachers. Write the ratio of teachers to
pupils.
Solution:
Number of teachers : Number of pupils
= 40 : 850 = 4:85
5. On a map, a distance of 5cm represent an actual distance of 15km. Write the
ratio of the scale of the map.
Solution:
Let x be the number to be added them
(49 + x) = (68 + x) = 3:4
4(49+X) = (68 + X)3
196+4X = 204 + 3X
4X – 3X = 204-196
X=8
EXERCISE 2.4.2
1. In the adjacent figure, two triangles are similar find the length of the
missing
side.
R
C
13cm
A
5cm
B
39cm
P
Solution:
Let the triangles be ABC and PQR AABC and APQR
= BC = AC
xcm
Q
QR
= 5
X
PR
= 13
39
13x = 5x39
2.
X = 5x39 = 5 x 3 = 15
13
What number is to 12 is 5 is to 30?
Solution:
Let x be the number
X: 12: : 5 : 30
30 x = 12 x 5
12x5
X = 30 = 2
3. Solve the following proportions:
(i) x : 5 = 3 : 6
(ii) 4 : y = 16 : 20
(iii)
2:3=y:9
(iv)
13 : 2 = 6.5 : x
(v) 2 : π = x : 22
7
Solution:
(i) x : 5 = 3 : 6
6x = 5x3
x = 5x3 = 2.5
6
(ii) 4 : y = 16 : 20
4x20 = 16y
y = 4x20 = 5
16
(iii)
2:3 = y:9
2x9 = 3y
Y = 2x9 = 6
3
(iv)
13:2 = 6.5 : x
13×x = 2x6.5
X = 2x6.5 = 1
13
(v)
2: π = x : 22
7
2x 22 = π x
7
X = 2x22 = 2
7
.π
( π = 22)
7
4. Find the mean proportion to:
(i)
8, 16
(ii) 0.3, 2.7
(iii) 16 2 , 6
3
(iv) 1.25, 0.45
Solution:
(i)
Let x to be mean proportion to 8 and 16
Then 8 = x
X 16
2
x = 8x16 = 128
x = √128 = √64x2 = 8 √2
(ii) Let x to be mean proportion to 0.3 and 2.7
Then 0.3 = x
X
2.7
X2 = 0.3x2.7 = 0.81
X = √0.81 = 0.9
(iii)
(iv)
Let the mean proportion to 16 2 and 6 be x.
3
Then 16 2 = x
3
6
X
2
X = 16 2 x6 = 50 x6 = 100
3
3
= x = √100 = 10
Let x be the mean proportion to 1.25 and 0.45.
Then 1.25 = x
X
0.45
2
X = 1.25x0.45
X = √1.25x0.45 =
1.25x45
√ 100x100
= 25x5x5x9 = 5x5x3 = 3
√ 100x100
10x10 4
5. Find the fourth proportion for the following:
(i)
2.8, 14, 3.5
(ii)
3 1 , 1 2, 2 1
3
(iii)
3
2
1 5, 2 3, 3 3
7
4
5
Solution:
(i)
Let x be the fourth proportion
Then 2.8 = 3.5
14
x
(ii)
2.8x = 14x3.5
X = 14x3.5 = 17.5
2.8
let x be the fourth proportion
31
Then
21
3 =
12
2
x
3
10
3 = 5
5
2
3
x
=
=
10 = 5
5
2x
10×2x = 5x5
(ii)
X = 5x5 = 5 = 1 1
10x2 4
4
Let x be the fourth proportion
15
7
= 3 3
2 3
14
5
x
12
7 = 18
31
5x
14
12 x 14 = 18
7
31 5x
12x2 = 18
31
5x
12x2x5x = 18x31
X = 18x31 = 93 = 4 13
12x2x5 20
6. Find the third proportion to:
(i)
12, 16
20
(ii) 4.5, 6 (iii) 5 1 , 16 1
2
2
Solution:
(i)
Let x be the 3rd proportion
Then 12 = 16
16 x
12x = 16x16
(ii)
X = 16x16 = 64
12
3
Let x be the third proportion to 4.5 and 6.
Then 4.5 = 6
6
x
4.5x = 6x6
X = 6x6 = 8
4.5
(iii)
Let x be the third proportion to 5 1 and 16 1
2
2
Then
51
2
16 1
16
=
2
11 33
2 = 2
33 x
2
1
2
x
11 = 33
33 2x
11x2x = 33x33
X = 33x33 = 99 = 49 1
11x2
2
2
7. In a map 1 cm represents 25km, if two cities are 2 1 cm apart on the map,
4
2
What is the actual distance between them?
Solution:
Let 2 1 cm represent x km
2
1 cm : 25 km: 2 1 cm : xkm
4
2
1 ×x = 25x2 1
4
2
X = 25x5
4
2
X = 25x5x4 = 250km
2
8. Suppose 30 out of 500 components for a computer were found defective.
At this rate how many defective components would he found in 1600
components?
Solution:
Number of defective components in 500 components = 30
Let x be the number of defective components in 1600 components
Then 30: 500 : : x: 1600
30x1600 = 500x
X = 30x1600 = 96
500
EXERCISE 2.4.3
1. Suppose A and B together can do a job in 12 days, while B alone can finish a
job in 24 days. In how many days can A alone finish the work?
Solution:
Number of days in which A and B together can finish the work = 12 days
Number of days in which B alone can finish the work = 30
1 =1 +1
T
m n
1 =1 +1
12 m 30
1 = 1 – 1 = 5-2 = 3 = 1
M 12 30 60 60 20
M = 20
i.e A can finish the work in 20 days.
2. Suppose A is twice as good a workman as B and together they can finish a
job in 24 days. How many days A alone takes to finish the job?
Solution:
A is twice as good a workman as B
i.e if B can finish a work in t days A can finish it in 1 days
2
1 =1 +1
T
m n
1 = 1 +1=2+ 1=3
24 t/2 t t t t
1 =3
24 t
t = 24x3 = 72
i.e. B takes 72 days to finish the job
A takes 72 = 36 days to finish it
2
3. Suppose B is 60% more efficient them A. if A can finish a job in 15 days how
many days B needs to finish the same job?
Solution:
A can finish a work in 15 days.
Work done A in 1 day = 1
15
B is 60% more efficient
Work done by B in 1 day
= 1 + 1 x 60
15 15 100
= 1 1 + 60
15
100
= 1 x8= 8
15 5 75
1
Number of days in which B alone can finish the work = 8 = 75
75 8
= 9 3 days
8
4. Suppose A can do a piece of work in 14 days while B can do it in 21 days.
They begin together and worked at it for 6 days. Then A fell ill B had to
complete the work alone. In how many days was the work completed?
Solution:
M = 14 days
N = 21 days
Part of work done in 6 days
=1 + 1 6
14 21
= 6 3+2 = 5 x6 = 5
42 42
7
Remaining part of the work = 1- 5 = 2
7 7
Days taken by B to finish
2
2 part of the work = 7 = 2x21 = 6 days
7
1 7 1
21
Total number of days in which the work is completed = 6+6 = 12 days
5. Suppose A takes twice as much time as B and thrice as much time as C to
complete a work. If all of them work together they can finish the work in 2
days. How much time B and C working together will take to finish it?
Solution:
If A alone takes to t1 days to do the work , B finishes it in t1
2
And C is t 1 days
3
1 =1 +1 +1
T
t1 t2 t 3
=1 + 1 + 1
T1
t1
t1
2
3
=1 +2 +3 = 6
T1 t 1 t1 t1
1 =1
T
2
1 =6
2
T1
That is, t1 = 12 days
i.e. B takes is 12 = 6 days
2
And C takes 12 = 4 days
3
Part of work done by B
In one day = 1
6
Part of work done by C in one day = 1
4
If B and C together takes t days to finish the work 1 = 1 + 1
T 6 4
= 2+3 = 5
12 12
T = 12 = 2.4 days
5
ADDITIONAL PROBLEMS ON PROPORTIONS
1. What number must be subtracted from each of the numbers 13, 17, 34, 42
so that the resulting four numbers are in proportion?
Solution:
Let x be the number to be subtracted from each
Then 13-x 17-x 34-x 42-x are in proportion
13 – x = 34 – 4
17 – x
42 - x
(13 – x) (42 – x) = (17 – x) (34 – x)
546 – 13x – 42x + x2 = 578 -17x – 34x + x2
-55x + 51x = 578 – 546
-4x = 32
X = 32 = -8
-4
i.e -8 must be subtracted from each number to get 4 numbers in proportion
2. Suppose A, B, C have a, b, c respectively if a:b = 4:5 b:c = 2:3 and a= 800
find C.
Solution :
A: b = 4: 5
B: c = 2: 3
i.e a: b = 8: 10
b: c = 10: 15
a: b: c = 8: 10: 15
a = 800
a: c :: 800: x where x is the amount C has 8 : 15 :: 800:x
8x = 15x800
X = 15x800 = 1,500
8
3. A boy 1.4 m tall casts a shadow 1.2 m long at the time when a building casts
a shadow 5.4m long . find the height of the building .
Solution:
A
p
B
C
1.2 cm
Q
5.4 m
Let x be the height of the building
Then 1.4 = 1.2
X
5.4
1.2x = 1.4x5.4
X = 1.4x5.4 = 14x54
1.2
12x10
= 63 = 6.3 m
10
R
4. if a = b c , what is a+b+c ?
3 4 7
c
Solution: let a = b = c = k
3 4 7
A = 3k, b= 4k, c = 7k
A+b+c = 3k + 4k + 7k = 14k = 2
C
7k
7k
5. If (a+b) : (a-b) = 1 : 5 find (a2 – b2) : (a2 + b2)
Solution: (a+b) : (a-b) = 1:5
5(a+b) = 1 (a+b)
5a + 5b = a-b
5a – a = -b -5b
4a = -6b
A = -6 = -3
B
4
2
i.e. a = -3k then b = 2k
a2 – b2 : a2 + b2 = (-3k)2 – (2k)2 : (-3k)2 + (2k)2
9k2 – 4k2 : 9k2 + 4k2
5k2 : 13k2
5 : 13
6. Arrange the ratios 7:20. 13 : 25, 17 : 30 and 11 : 15 in decreasing order.
Solution : the ratio are 7 , 13 , 17 and 11
20 25 30
15
LCM = 5x2x3x2x5
= 300
7 = 105
20 300
13 = 156
25 300
17 = 170
30 300
11 = 220
15 300
105<156<170<220
i.e the ratios in the decreasing order are,
11:15, 17:30, 13:25 and 7:20
7. The cost of making an article is divided between materials, labour and
overheads in the ratio 5:3:1 . the cost of the materials is 125, what is the
cost of labour?
Solution :
Ratio of cost of materials, labour and over heads = m: L :0 = 5:3:1
Cost of materials = 125.00
Let m = 5k L = 3k and 0 = k
5k = 125
K = 125 = 25
5
Cost of labour = 3k = 3x25 = 75.00
8. A port had provisions for 450 men for 80 days. After 10 days 50 more men
arrived. How long will the remaining food last at the same rate?
Solution:
When 50 more man joined there was provisions for 450 men for (80-10) =
70 days
We have to find out, for how many days the provisions will last if the
number of men was 450+50 = 500
Men days
450 70
500
?
= 70x450 = 63 days
500
9. A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern
fills up in 10 hours. If the cistern is full, in how much time will be emptied
by rhe leak?
Solution:
Then taken by the pipe to fill the cistern = 9 hours
Part filled (when there is no leake) in 1 hour = 1
9
Let the full insterm empty in x houses
Part empted per hour = 1
X
Part filled in 1 hour = 1 - 1
9 x
Time taken to fill the tank
When the leak is not plugged = 10 hours
Part filled per hour = 1
10
1 - 1 = 1
9 x
10
-1 = 1 - 1 = 9-10 = -1
X 10 9
90
90
1 =1
X 90
X = 90
The cistern will empty in 90 hours.