|
|
Exchange Rates | Simple Interest | Compound Interest | Speed | Density | Triangle Method
|
|
Exchange Rates
This is a practical use of direct proportion.
Example 1: Given £1 = €1.5, change £10 into euros.
Example 2: Change €28 into pounds.
| |
€ 1 = £1/1.5 |
| |
€28 = 1/1.5 x 28 = £18.67 (to the nearest penny) |
Rule: Work out what one unit is in the money you are changing. Then multiply by the amount you are changing.
|
Simple Interest (I)
This is a practical use of percentages. It is a way of working out the interest gained on an investment.
| The formula for simple interest is: |
I = PTR
100
|
|
| |
|
|
where
|
|
|
| P = Principal (amount of money invested) |
| T = Time (in years) |
|
|
| R = Rate of interest (%) |
|
|
Example 1: £600 is invested for 8 years at 15% simple interest. How much interest will it earn? How much will the amount be after 8 years?
|
|
|
|
|
|
|
|
I =
|
|
600 x 8 x 15
100 |
|
R = 15
|
|
|
|
|
| |
I =
|
|
£720
|
|
| |
Amount =
|
|
600 + 720 =
|
£1320 |
Note: questions on this topic can ask for I, P, T, or R.
Simply make the required term the subject of the formula and substitute in the values given.
Example 2: £6,000 was invested for one year and six months. If the simple interest gained was £1,000, what was the rate of interest?
|
|
|
|
|
|
|
|
R =
|
|
100 I
PT
|
(making R the subject) |
R = ?
|
|
|
|
|
I = 1000
|
|
|
|
|
| |
R =
|
|
100 x 1000
|
= 11
|
1/9 |
| |
|
|
6000 x 1½
|
|
| |
R=
|
|
11 |
1/9 % |
|
|
|
Compound Interest
In this case the interest gained is added on each year and the next year's interest worked out.
Example 1: Work out the compound interest on £900 invested for 3 years at 5%.
| |
1st year 5/100 x 900 = £45
2nd year 5/100 x 945 = £47.25
3rd year 5/100 x 992.25 = £49.61 (to nearest penny)
|
Total compound interest = 45 + 47.25 + 49.61 = £141.86
Example 2: Calculate the amount after 3 years
| |
Amount = 900 + 141.86 = £1,041.86 |
For a larger number of years, we use a shorter method.
Using the figures in this example:
The amount at the end of the first year = 105% of 900 = 1.05 x 900
At the end of the second year = 105% of the first year = 1.05 x 1.05 x 900
So each year we multiply by 105% = 1.05
Final amount = (1.05)3 x 900 = £1,041.86
(Note: this can be done on the calculator as follows)
|
1.05 (power button) 3 x 900
^
|
For 15 years, we simply change the power to 15.
(1.05)15 x 900 = £1,871.04 (nearest penny)
|
Speed
Speed is defined as the distance travelled in one unit of time.
Example 1: A car travels 150 km in two hours. What was its average speed?
| |
In 1 hour the car travels 150 ÷ 2 = 75 km |
| |
Speed = 75 km/hr
|
The units of speed can be km/hr or m/sec.
Note: always use the units asked for in a question.
Example 2: A particle travels 30000 cm in 0.5 mins. What is its speed in m/sec?
30000 cm =
0.5 mins =
Speed =
|
300 m
30 secs.
300/30 = 10 m/sec
|
|
Formula for average speed (S)
|
S = D
T
Where S = Speed
D = Distance
T = Time
|
To use this formula two of the values are given. By changing the subject of the formula, we can calculate the third.
Example 3: A car travels at a speed of 80 km/hr for 2 hours. What distance did it travel?
S = 80
T = 2
D = S x T (making D the subject)
D = 80 x 2 = 160 km
Example 4: A car travels 300 km at a speed of 60 km/hr. How long did it take?
| |
|
|
| S = |
60 |
|
| D = |
300 |
|
| T = |
D |
|
| |
S (making T the subject)
|
|
| T = |
300 = 5 hours |
|
| |
60 |
|
|
Density
Density is the mass per unit volume.
|
Formula for Density (D)
|
|
D = M
|
|
V
|
|
|
Where M = mass (g)
V = volume (cm3)
|
Example 1: The mass of a solid is 500g and the volume is 1000 cm3. Calculate the density.
M = 500
V = 1000
D = 500 = 0.5 g/cm3
1000
Example 2: V = 3000cm3, D = 0.2 g/cm3, so M = ?
M = D x V (making M the subject)
M = 0.2 x 3000 = 600g
|
|
Triangle Method
Both types of problems relating to speed and density can be solved by using a triangle. This is known as the Triangle Method for Speed and Density.
Speed
The vertical line in the triangle means multiply and the horizontal line means divide.
So if two of the values of D, T or S are known, then you can work out the other using one of the following:
 |
|
D/T = S
D/S = T
S x T = D
|
|
Density
The triangle for density is used in the same way.
 |
|
M/D = V
M/V = D
D x V = M
|
|
|
