Sect 5.1 - Exponents: Multiplying And Dividing Common Bases Worksheet With Answers Page 6
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1
2
3
4
5
6
745
i = prt = (1144.90)(0.07)(1) ≈ $80.14.
The total in the account is
A = p + i = 1144.90 + 80.14 = $1225.04
At the end of three years, Juan has $1225.04 in his account.
This type of interest computation in called compound interest . Not only is
one earning interest on the principal, but also earning interest on the
previous interest earned. To see how this works, let us examine the
previous example more closely.
For the first year, Juan’s interest was $70 = 1000(0.07) and the total he
had was $1070 = 1000 + 70 = 1000 + 1000(0.07) = 1000(1 + 0.07).
Hence, $1070 = $1000(1 + 0.07) .
For the second year, Juan’s interest was $74.90 = 1070(0.07) and the total
he had was $1144.90 = 1070 + 74.90 = 1070 + 1070(0.07)
= 1070(1 + 0.07) .
Hence, $1144.90 = $1070(1 + 0.07). But, $1070 = $1000(1 + 0.07) , so we
can substitute $1000(1 + 0.07) in for $1070:
$1144.90 = $1070(1 + 0.07).
2
$1144.90 = $1000(1 + 0.07) (1 + 0.07) = $1000(1 + 0.07)
2
Thus, $1144.90 = $1000(1 + 0.07)
For the third year, Juan’s interest was $80.14 = 1144.90(0.07) and the total
he had was $1225.04 = 1144.90 + 80.14= 1144.90 + 1144.90(0.07)
= 1144.90(1 + 0.07) .
2
Hence, $1225.04 = $1144.90 (1 + 0.07). But, $1144.90 = $1000(1 + 0.07)
,
2
so we can substitute $1000(1 + 0.07)
in for $1144.90 :
$1225.04 = $1144.90 (1 + 0.07)
2
3
$1225.04 = $1000(1 + 0.07)
(1 + 0.07) = $1000(1 + 0.07)
3
Hence, $1225.04 = $1000(1 + 0.07)
.
Notice that $1225.04 was the amount, $1000 was the original principal,
0.07 was the rate as a decimal, and the exponent 3 was the time in years.
From this, we can write a formula for interest compounded annually:
3
$1225.04 = $1000(1 + 0.07)
.
↓
↓
↓ ↓
t
A
=
P (1 +
r )
Let us verify that the formula gives us the correct answer:
3
A = 1000(1 + 0.07)
(#1-parenthesis, # 4-add)
3
= 1000(1.07)
(#2-exponents)
= 1000(1.225043)
(#3-multiply)
= 1225.043 ≈ $1225.04 which matches the answer to #10.
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