Strand 3 — Number
Ratio, Proportion, and Percentages
1st Year · 2nd Year · 3rd Year (Junior Cert)
- ✓By the end of this lesson students will be able to express quantities as ratios and simplify them.
- ✓By the end of this lesson students will be able to distinguish between direct and inverse proportion and solve related problems.
- ✓By the end of this lesson students will be able to calculate percentage increase and decrease, and find the original amount.
- ✓By the end of this lesson students will be able to apply the compound interest formula to calculate final amounts and interest earned over multiple periods.
Key concepts
A ratio is a comparison of two or more quantities of the same unit. It shows how much of one quantity there is compared to another. Ratios can be written in the form a:b or as a fraction a/b. Ratios should always be simplified to their lowest terms.
Proportion describes how two quantities are related. If two ratios are equal, they are in proportion. There are two main types: direct proportion and inverse proportion.
Two quantities are in direct proportion if an increase in one quantity leads to a proportional increase in the other, and a decrease in one leads to a proportional decrease in the other. Their ratio remains constant. If y is directly proportional to x, then y = kx, where k is the constant of proportionality.
Two quantities are in inverse proportion if an increase in one quantity leads to a proportional decrease in the other, and vice versa. Their product remains constant. If y is inversely proportional to x, then y = k/x, where k is the constant of proportionality.
A percentage is a way of expressing a number as a fraction of 100. The word 'percent' means 'per hundred'. It is denoted by the symbol '%'. To convert a percentage to a decimal, divide by 100. To convert a decimal or fraction to a percentage, multiply by 100.
To calculate a percentage increase, add the percentage amount to the original value. To calculate a percentage decrease, subtract the percentage amount from the original value. A quicker method is to use a multiplier: for an x% increase, multiply by (1 + x/100); for an x% decrease, multiply by (1 - x/100).
Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. It's essentially 'interest on interest'. It leads to faster growth of an investment or debt compared to simple interest. The formula calculates the final amount (F) after a certain time (t) when a principal amount (P) is invested at an annual interest rate (i) compounded annually.
Key facts to remember
- 1Ratios compare quantities of the same unit and should always be simplified.
- 2Direct proportion: y = kx (as one quantity increases, the other increases proportionally).
- 3Inverse proportion: y = k/x (as one quantity increases, the other decreases proportionally).
- 4To convert a percentage to a decimal, divide by 100 (e.g., 25% = 0.25).
- 5To find a percentage increase, multiply by (1 + decimal percentage).
- 6To find a percentage decrease, multiply by (1 - decimal percentage).
- 7Compound interest formula: F = P(1 + i)^t, where F is final amount, P is principal, i is interest rate (as decimal), and t is time.
- 8Always round currency answers to two decimal places.
Worked examples
Example 1
a) If 5 workers can paint a house in 12 days, how many days would it take 3 workers to paint the same house, assuming they work at the same rate? b) A recipe requires 200g of flour for 8 scones. How much flour is needed for 12 scones?
Answer
a) It would take 3 workers 20 days to paint the house. b) 300g of flour is needed for 12 scones.
Always identify if it's direct or inverse proportion before solving.
Example 2
A jacket costing €80 is reduced by 25% in a sale. What is the new price of the jacket? If the sale price is then increased by 10% for the new season, what is the final price?
Answer
The new price after the reduction is €60. The final price after the increase is €66.
Be careful to apply the second percentage change to the *new* price, not the original price.
Example 3
Aoife invests €2,500 in a savings account that offers a compound interest rate of 3% per annum. Calculate the total amount in her account after 3 years, assuming no withdrawals or further deposits.
Answer
The total amount in Aoife's account after 3 years will be €2,731.82.
Remember to convert the percentage interest rate to a decimal before using it in the formula.
Common mistakes
- ✗Confusing direct and inverse proportion, leading to incorrect calculations.
- ✗Calculating a percentage increase/decrease on the original amount instead of the new amount after a previous change.
- ✗Forgetting to convert the interest rate from a percentage to a decimal in the compound interest formula.
- ✗Incorrectly applying the order of operations (BIDMAS/BODMAS) when calculating compound interest, especially with the exponent.
- ✗Not simplifying ratios to their lowest terms.
Exam tips
- ★Read the question carefully to determine if it's direct or inverse proportion.
- ★When dealing with multiple percentage changes, apply them sequentially to the updated amount.
- ★Show all steps in your calculations, especially for compound interest, to gain full marks even if there's a minor arithmetic error.
- ★Always check your answer for reasonableness. Does a 25% reduction on €80 make sense as €60? Yes.
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