Ratio
A ratio is the quotient of two quantities of the same kind. In a to b, a is the antecedent and b is the consequent. A ratio has no unit. Multiplying both terms by the same non-zero number keeps the ratio the same.
The compound ratio of a to b and c to d is a times c to b times d. For a continued ratio, make the common term equal. If a to b is 2 to 3 and b to c is 4 to 5, then a to b to c is 8 to 12 to 15. To divide an amount A in the ratio x to y, the first share is A times x over x plus y, and the second is A times y over x plus y.
Proportion
Four quantities a, b, c and d are in proportion if a to b equals c to d. Then a times d equals b times c. In direct proportion, one quantity rises when the other rises. In inverse proportion, one rises when the other falls, and their product stays fixed.
Partnership
Profit is shared in the ratio of capital times time. If capitals are equal, use the ratio of times, and if times are equal, use the ratio of capitals.
Simple Interest
With principal P, rate r percent per year and time t years, simple interest equals P times r times t divided by 100. The amount is P plus the interest.
Compound Interest
The amount equals P times one plus r over 100, raised to the power n. For half-yearly compounding, halve the rate and double the number of periods. For two years, compound interest minus simple interest equals P times r over 100, squared.
Shortcuts
If money doubles in n years at simple interest, the rate is 100 divided by n percent. To become k times, the time is k minus one, times 100, divided by r years. At 10 percent, the two-year compound multiplier is 1.21 and the three-year multiplier is 1.331.
Worked Examples
Example 1: Find the simple interest on 5000 taka at 8 percent for 3 years.
Solution: 5000 times 8 times 3 divided by 100 equals 1200 taka. The amount is 6200 taka.
Example 2: Divide 4200 taka in the ratio 3 to 4.
Solution: The total parts are 7. The first share is 4200 times 3 over 7, which is 1800. The second is 2400.
Example 3: If a to b is 2 to 3 and b to c is 4 to 5, find a to b to c.
Solution: Make b equal to 12. Then a to b to c is 8 to 12 to 15.
Quick Check
Question 1: What is the formula for simple interest?
Answer: Principal times rate times time divided by 100.
Question 2: What is the difference between compound and simple interest for two years?
Answer: P times r over 100, squared.
Question 3: If money doubles in 8 years at simple interest, what is the rate?
Answer: 12.5 percent.