A new machine is bought for €30000 - Leaving Cert Mathematics - Question 1 - 2017

To find the initial sum invested, we use the formula for compound interest:

$A = P(1 + r)^t$
where:

• $A = 30000$
• $r = 0.03$ (3% interest)
• $t = 2$ (duration in years).

Rearranging gives us:

$P = \frac{A}{(1 + r)^t}$
Substituting the values:
$P = \frac{30000}{(1 + 0.03)^2}$
Calculating this value:

• $(1 + 0.03)^2 = 1.0609$
• Therefore,
  $P \approx \frac{30000}{1.0609} \approx 28277.88$

Thus, the sum invested was approximately €28,277.88.

To find the rate of interest, we also use the compound interest formula, rearranging it. The formula is given by:

$A = P(1 + i)^t$
To solve for $i$, we need to rearrange:

$1 + i = \left( \frac{A}{P} \right)^{\frac{1}{t}}$
So,
$i = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1$
Substituting in our values:

• $A = 26530.20$
• $P = 25000$
• $t = 3$
  We find:
  $i = \left( \frac{26530.20}{25000} \right)^{\frac{1}{3}} - 1$
  Calculating gives us:
• $\frac{26530.20}{25000} \approx 1.061208$
• Taking the cube root:
  $(1.061208)^{\frac{1}{3}} \approx 1.02$
  Thus, $i \approx 0.02$, which translates to a rate of interest of approximately 2%.