Question

A man gets a Rise in 10% of salary at the end of this first year of service and the further rises of 20% or 25% at the end of second year and third year respectively. To rise in each Case is being calculated on his salary on the beginning of the year. To what annual percentage increase in this equivalent.

Answer 1

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Assume his salary starts at 1.

at the beginning of the first year he is making 1.

at the end of the first year he is making 1 * 1.1 = 1.1

at the end of the second year he is making 1.1 * 1.2 = 1.32

at the end of the third year he is making 1.32 * 1.25 = 1.65

he starts at 1 and at the end of the third year he is making 1.65

his average annual increase is given by the compounding formula of:

• 1.65 = 1*(1+x)^3
• this is equivalent to:
• 1.65 = (1+x)^3
• take the cube root of both sides to get:
• (1.65)^(1/3) = 1+x
• subtract 1 from both sides to get:
• x = (1.65)^(1/3) - 1
• solve for x to get:
• x = .18166575
• at the beginning of the first year he is making 1.

at the end of the first year he is making 1 * 1.18166575 = 1.18166575

at the end of the second year he is making 1.18166575 * 1.18166575 = 1.396333946

at the endof the third year he is making 1.396333946 * 1.18166575 = 165

1 * (1.18166575)^3 = 1.65

the equation that was used was the future value of a present amount formula that is equal to:

• f = p * (1+i)^n
• f = future value
• p = present amount
• i = interest rate per time period
• n = number of time periods.

to solve this problem, we first had to find f.

that was done using the year by year analysis up top.

once we knew f, we could then substitute in the formula to get:

f = 1.65

p = 1

i = x

n = 3

we then solved for x

Answer 2

Answer:

his salary starts at 1.

first year he is making 1.

first year - 1 * 1.1 = 1.1

second year - 1.1 * 1.2 = 1.32

third year - 1.32 * 1.25 = 1.65

starts at 1 and a end of the third year - 1.65

his average annual increase is given by the compounding formula of:

1.65 = 1*(1+x)^3

this is equivalent to:

1.65 = (1+x)^3

take the cube root of both sides to get:

(1.65)^(1/3) = 1+x

subtract 1 from both sides to get:

x = (1.65)^(1/3) - 1

solve for x to get:

x = .18166575

end of the first year - 1 * 1.18166575 = 1.18166575 second year - 1.18166575 * 1.18166575 = 1.396333946

third year - 1.396333946 * 1.18166575 = 165

1 * (1.18166575)^3 = 1.65

the equation that was used was the future value of a present amount formula that is equal to:

f = p * (1+i)^n

f = future value

p = present amount

i = interest rate per time period

n = number of time periods.

to solve this problem, we first had to find f.

once we knew f, we could then substitute in the formula to get:

f = 1.65

p = 1

i = x

n = 3

we then solved for x.