1. Let the required sum be Rs. P. Then

Rs. P = Rs. 100 143 100 143 4 2

Rs.1 1 13 5

3 24 2

= Rs. 1760.

To find rate %

Since I = prt100

∴ r =100Ipt

2. Here P = Rs. 468.75, t = 2 5

1 or 3 3

years.

I = Rs. (500 − 468.75) = Rs. 31.25

∴rate p.c. = 100 31.25 3125 3

1005 46875 5

468.7453

= 4

3. Rs. 600 f or 2 years = Rs. 1200 for 1 years

And Rs. 150 for 4 years = Rs. 600 for 1 year

Int. = Rs. 90.

∴Rate = 90 1001800 1

= 5 %

To find Time

Since I = prt100

. ∴ t = 100I

Pr

4. Here interest = Rs. 15767.50 − Rs. 8500 = Rs. 7267.50

∴ t = 7267.50 1008500 4.5

= 19 years.

5. Let Principal = P, time = t ears, rate = t

Then P t t100 9

P

∴ t 2 =

100

9 ∴t =

10 13

9 3 ∴ rate =

13 %

3

Race Formula

. Rate = time = 1 10 1

100 3 %9 3 3

6. Let the annual payment be P rupees.

The amount of Rs. P in 4 years at 5%

The amount of Rs. P in 4 years at 5% = 100 4 5 120

100 100P P

“ “ “ 3 “ = 115P100

“ “ “ 2 “ = 110P100

“ “ “ 1 “ = 105P100

These four amounts together with the last annual payment of Rs. P will discharge the debt of

Rs. 770

∴115 P 110 P 105 P120 P

P100 100 100 100

= 770

∴550 P

100= 770 ∴

770×100

550= 140

Hence annual pay ment = Rs. 140

Theorem: The annual payment that will discharge a debt of Rs. A due in t years at the rate of interest r % per

annum is 100

1002

AA

rt t At

Proof: Let the annual payment be x rupees.

The amount of Rs. x in (t − 1) yrs at r % = 100 1

x100

t r

The amount of Rs. x in (t − 2) yrs at r % =100 2

x100

t r

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The amount of Rs. x in 2 yrs at r % = 100 2

x100

r

The amount of Rs. x in 1 yrs at r % = 100 1

x100

r

These (t −1) amounts together with the last annual pay ment of Rs. X will discharge the debt of Rs. A

∴100 1 100 2 100

x + x ... + x + x = A100 100 100

t r t r r

Or, x [{100 + (t − 1) r} + {100 + (t − 1) r} + ... {100 + r}]

= 100A

Or, x 1

100 1002

r t tA ∴x =

100

1100

2

A

r t tt

Note: 1 + 2 + 3 + ... + m = 1

2

m m

Using the above theorem:

Annual payment = 100 770 770 100

5505 4 55 100

2

= Rs. 140

7. By theorem payment = 848 100

4 3 44 100

2

= Rs. 200

8. Putting the values in the above formula:

80 = 100

5 4 55 100

2

A

Or, A = 80 550

100= Rs. 440

9. Let his deposit = Rs. 100

Interest for first 2 yrs. = Rs. 6

Interest for next 3 yrs. = Rs. 24

Interest for the last year. = Rs. 10

Total interest = Rs. 40

When interest is Rs. 40, deposited amount is Rs. 100

∴ when interest is Rs. 1520, deposited amount

= 100

152040

= Rs. 3800

Race formula:

Principal = + ...

1 1 2 2 3 3

1520 × 100Intereset ×100t r + t r + t r 2 × 3 + 3 × 8 + 1 × 10

= 1520 × 100

40= Rs. 3800

10. Let the sum it become Rs. 200.

∴ Interest = 200 − 100 = 100

Then, rate = 100 I

Pt=

100 ×100100×10

= 10%

Race formula:

Time × Rate = 100 (Multiple number of principal − 1)

Or, Rate = 100 × Multiple number of principal 1

time

Using the above formula: rate = 100 2 1

10= 10%

11. Rate = 100 3 1

20= 10%

Note: A generalized form can be shown as:

If a sum of money becomes ‘x’ times in‘t’ years as SI, the rate of interest rate is given by

100 x 1

%t

12. Using the above formula.

Time = 100 Multiple number of principal 1

Rate

= 100 4 1

5= 60 years.

13. Amount of 1st part =

1101st part

100

“ “ 2nd

part = 115

2nd part100

“ “ 3rd

part = 120

3rd part100

According to t he question, these amounts are equal

∴ 110 × 1st part = 115 × 2

nd part = 120 × 3

rd part

∴ 1sr part: 2nd

part: 3rd

par t=1 1 1

:110 115 120

:

Hence, dividing Rs. 2379 into three parts in the ratio 276: 264: 253, we have 1st part = Rs. 828,

2nd

part= Rs. 792, 3rd

part = Rs. 759.

14. P + SI for 3.5 yrs = Rs. 873

P + SI for 2 yrs = Rs. 756

On subtracting, SI for 1.5 yrs = Rs. 117

Therefore, SI f or 2 yrs = Rs. 117

21.5

= Rs. 156

∴ P = 756 − 156 = Rs. 600

And rate = 100 156600 2

= 13% per annum

15. Let the sum be Rs. x and the original rate be y% per annum. Then, new rate = (y + 3) % per annum.

∴3 2 2

100 100

x y x y= 300

xy + 3x − xy = 15000 or x = 5000

Thus, the sum = Rs. 5000

Race formula: Direct Formula

Sum = More Intereset × 100 300 100

2 3Time × More rate= 5000

16. Rate = 100(2 1) 100

7 7

100(4 121

100

7

Time years

OTHER METHOD: This question can be solved without writing anything.

Think like.

Doubles in 7 years

Trebles in 14 years

4 times in 21 years

5 times in 28 years

And so on.

17. Let the amount lent at 3% rate be Rs x, then

3% of x + 5% of (4000 – x) = 144

Or, 3x + 5 4000 – 5x = 14400

Or, 2x = 5600 X = 2800

Thus, the two amounts are Rs 2800 and (4000–2800) or Rs 1200

18. First rate of interest = 120 100

800 3

= 5%

New rate = 5 + 3 = 8%

New interest = 800 3 8

192.100

Rs

New amount = 800 + 192 = 992.

19. Simple interest for 5 years = Rs 300

Now, when principal is trebled, the simple interest for 5 years will also be treble the simple interest

On original principal for the same period. Thus SI for last 5 years when principal is trebled

=3 300 = Rs 900

Total SI for 10 yrs = 300 + 900 = Rs 1200

Theorem: A sum of Rs X is lent out in n parts in such a way that the interest on first part at r1% for t1

yrs, the interest on

Second part at r2% for t 2 yrs the interest on third part at r3% for t3 yrs. And so on, are equal, the ratio

in which the sum was divided in n parts is given by1 1 2 2 3 3

1 1 1 1: : : ............

n nr t r t r t r t.

Proof: Let the sum be divided into 1 2, ............ nS S S

1

2

3

int 100

1 1

int 100

2 2

int 100

3 3

int 100n

n n

Sr t

Sr t

Sr t

Sr t

[Since interest of all parts are equal]

1 2 3

int 100 int 100 int 100 int 100: : : ........ : : ................. :

1 1 2 2 3 3n

n n

S S S Sr t r t r t r t

= 1 1 2 2 3 3

1 1 1 1: : : ............

n nr t r t r t r t

20. Each Interest = 1 5 10 2 6 9

100 100

stpart ndpart

Or,

1 6 9 2727 : 25

2 5 10 25

26001 27 1350

27 25

2 2600 1350 1250

stpart

ndpart

stpart Rs

and ndpart Rs

21. Interest = Rs 840 – Rs 750 = 90

Time = 90 100

3750 4

yrs

Now, by the formula:

Sum = 100 100 575

500100 100 3 5

amountRs

rt

Note: There is a direct relationship between the principal and the amount

And is given by 100

100

AmountSum

rt

22. Use the formula

Principal = 100 100 2613 100 2613

2010100 100 30 130

AmountSum Rs

rt

Again by using the same formula:

100 30152010

100 5

100 3015,100 5

2010

1 100 3015 100 2010

5 2010

100 (3015 2010) 100 1005 100 100510

2010 5 2010 5 2010 5

t

or t

t

years

23. Let the sum be Rs X.

Interest = 8 4 32

100 100

x x

32 68

100 100

x xx

When interest is 68

100

x less, the sum is Rs X.

When interest is 340less, the sum is 100 340 50068

xRs

x

Direct formula: 100 100 340

340 500100 8 4 68

Sum Rs

24. We may consider that Rs (1800 – 1650) gives interest of Rs 30 at 4% per annum.

30 100

5 .150 4

Time yrs

25. After 2 yrs amount returned to Ramu

400 5 2

400 440100

Rs

Amount returned to Arun = 2% of Rs 440 = Rs 8.80.

26. Theorem: When different amounts mature to the same amount at simple rate of interest, the ratio of

the amounts

Invested are in inverse ratio of (100 + time rate). That is, the ratio in which the amounts are

invested is

1 1 2 2 3 3

1 1 1 1: : : .................. :

100 100 100 100 n nr t r t r t r t

Proof: We know that sum = 100

100

amount

rt

Let the sum invested be1 2, ............ nS S S ,

At the rate of 1 2 3, , .................. nr r r r for time

1 2 3, , , nt t t t yrs respectively Then

1 2 3: : : .................. : nS S S S

1 1 2 2 3 3

100 100 100 100: : : ....................... :

100 100 100 100 n n

A A A A

rt r t r t r t

[Since the amount (A) is the same for all]

1 1 2 2 3 3

1 1 1 1: : : ....................... :

100 100 100 100 n nr t r t r t r t

Soln: Therefore, the required ratio is this case is

1 1 1 1 1 1

: : : :100 2 5 100 3 5 100 4 5 110 115 120

27. Doubles in 4 yrs

3 times 4 2 = 8 yrs

4 times 4 3 = 12 yrs

8 times 4 7 = 28 yrs

Thus direct formula: X Times in = No. of yrs to double (X – 1)

8 times = 4(8 – 1) = 4 7 = 28 yrs.

28. Let the sum be Rs X, then

15 5 15 7144

100 200

,150 105 144 200

144 200640

45

x x

or x x

x Rs

Direct formula: Two equal amounts of money are deposited at r1% and r2% for t1 and t2 yrs

respectively. If the difference between

Their interest is di then sum = 1 1 2 2

100di

r t r t

Thus, in this case: sum = 144 100 144 100

64015 5 15 3.5 22.5

Rs

29.

1

1 1

2

2 2

1 2 1 2

500 210

100

500 210

100

10 10 2.5

rI r

rI r

I I r r

Or, r1–r2 = 2.5

10 = 0.25%

By Direct formula: When t1 = t2,

(R1–r2) =100 2.5 100

0.25%500 2

dI

Sum t

30. Let the sum be Rs X, then at 4% rate for 4 yrs the simple interest

= 4 4 4

100 25

x xRs

At 5% rate for 3 yrs the simple = 5 3 3

100 20

x xRs

Now, we have,

4 380

25 20

16 15, 80

100

8000

x x

x xor

X Rs

Quicker Method: For this type of question

Sum = 1 1 2 2

100 80 100

4 4 3 5

Difference

r t r t

= Rs 8000

31. Suppose the of interest = r% and the sum = Rs A

Now, A + 4

600;100

A r or, A + 600

25

rA

, 1 600 (1)

25

6, 650;

100

ror A

A rand A

, 1 600

25

6, 650; (2)

100

ror A

A rand A

Dividing (1) by (2), we have

1

600 (25 ) 2 1225 ; ,3 650 50 3 13

150

r

rOr

R r

Or, (50 + 2r) 13 = (50 + 3r) 12

Or, 650 + 26r = 600 + 3r; or, 10r = 50 r = 5%

Direct formula: If a sum amounts to Rs A1 in t1 years and Rs A2 in t2 years at simple rate of interest,

Then rate per annum =

2 1

1 2 2 1

100 A A

A t A t

In the above case,

R = 100 650 600 100 50

5%6 600 4 650 1000

32. Any sum that is paid back to the bank before the last instalment is deducted from the principal and not

from the interest.

Total interest = Interest on Rs 7000 for 3 yrs + Interest on (Rs 7000 – Rs 3000) = Rs 4000 for 2 yrs.

Or, (5450 + 3000 – 7000) = 7000 3 4000 2

100 100

r r

Or, 1450 = 210r + 80r 1450

290r = 5%

33. Suppose Rs X was lent at 6% per annum.

Thus, 6 5 (7000 ) 4 5

1600100 100

x x

3 7000

, 160010 5

x xor

3 14,000 2

, 160010

x xor

X = 16000 – 14000 = Rs 2000.

By Method of Alligation: Overall rate of Interest = 1600 100 32

%5 7000 7

Ratio of two amount = 2: 5

Amount lent at 6% = 7000

27

=Rs 2000

34. SI = 400 5 6

100

= Rs 120

35. Interest = Rs. 1 146 15 1

3064 365 4 100

= Rs. 1225 2 15 1

4 5 4 100

= Rs. 147

32= Rs. 4.59 nearly.

6%

32%

7

4%

10%

7

4%

7