Gasoline prices in Matthews town have increases 18 each year

Gasoline prices in Matthew\'s town have increases 18% each year for the past two years. Let g be the current price of a gallon of gas in Matthew\'s town, Write an equation that describes the projected price per gallon ,p ,at a time t years from now if this trend continues. Rewrite your equation to find the effective quarterly percentage increase, to the nearest hundredth of the percent.

Solution

a) Current Prices of Gasoline: g

Increase Percentage, r   = 18% = 18/100

Total number of Years = t

Projected Price: p

Let Suppose,

If, t = 1 then,

p = g

If, t = 2 then,

P = First Year price + Increased Price

                = g + 18 percent of g    = g + 18g/100   = g (1 + r)

If, t = 3 then,

P = 2^nd Year price + Increased Price from 2^nd Year

   = > p = g (1 + r ) + r * g (1+r) = g (1+r) (1+r) = g (1+r)^2

If, t = 4 then,

3^rd Year price + Increased Price from 3^rd Year

If, t = t then we get,

P = g (1+r) ^ (t-1)

Where r = 18/100 = 18%

So, p = g (1.18) ^t-1

B )

For Quarterly increase,

We have time period increase and decrease in Rate of interest.

In a year we have 4 quarters, so rate is divided by 4 and time is multiplied by 4

Hence, in the above equation, t = 4t and r = r/4, we get the final result as

P (inc) = g (1 + r/4) ^ 4t-1 = g (1 + 18/400) ^ 4t – 1   = g (418/400) ^ 4t-1

P (inc) = g (418/400) ^ 4t-1

Percentage Increase = (Increase – original ) / Original Percent

= [ g (418/400) ^4t-1 - g (1.18) ^ t-1 ] / g(1.18) ^ t-1   * 100

= g {((1.045)^4)^t / 1.045 – 1.18^t / 1.18} / g (1.18)^t-1 * 100

= [ ( 1+r/4) ^ 4t / (1 + r/4) / (1+r) ^ t / (1+r)    - 1 ] * 100

= { (1 + r/4)^4t-1 / (1+r)^t-1 } - 1 Percent

= [ { (1 + r/4)^4t-1 / (1+r)^t-1 } - 1 ] * 100  

Answer for second Question