Ratio, VAT, interest and speed

Ratio and proportion

A ratio compares quantities of the same kind. Aoife and Conor share a bag of sweets in the ratio $2 : 3$. For every 2 sweets Aoife gets, Conor gets 3.

To simplify a ratio, divide both parts by the same number (their HCF), just as you simplify a fraction. For example, $18 : 24 = 3 : 4$ (divide both by 6).

To divide an amount in a ratio, add the parts, find the value of one part, then multiply.

When two ratios are equal, you have a proportion. To find a missing value, look for the number that multiplies one side.

Percentages and VAT

VAT (value-added tax) is a tax added to the price of most goods and services. The standard rate in Ireland is $23\%$, and a reduced rate of $13.5\%$ applies to some things.

The quick way is to use a multiplier. Adding $23\%$ means you pay $100\% + 23\% = 123\%$ of the price, so multiply by $1.23$. Taking off $15\%$ means you pay $85\%$, so multiply by $0.85$.

Going backwards

If you know the price including VAT, that price is $123\%$ of the original. To get back, divide by $1.23$.

A useful fact for sales: a price multiplied by $1.23$ and then by $0.85$ gives the same result as multiplied by $0.85$ and then by $1.23$, because the order of multiplying does not matter.

Compound interest

When you put money in a savings account, the bank pays you interest. The amount you start with is the principal, $P$.

With simple interest, you earn interest only on the principal each year. With compound interest, interest is added to the account each year, so next year you earn interest on the larger amount as well.

If the rate is $i$ (as a decimal) per year, the account is multiplied by $(1 + i)$ every year. After $t$ years:

You can also check year by year: $2000 \to 2080 \to 2163.20 \to 2249.728$.

Finding the rate or the principal

Between consecutive years the account is multiplied by $(1 + i)$. So divide one year's value by the year before to get the multiplier. To get back to the start, divide by the multiplier.

Pay and tax

When you get a job, your gross pay is what you earn before anything is taken off. Net pay is what you actually take home.

Income tax in Ireland is charged at two rates. The standard rate (here $20\%$) applies up to the standard rate cut-off point. The higher rate (here $40\%$) applies to everything above it. Then tax credits are subtracted from the tax, not from the pay. USC and PRSI are other deductions.

Currency conversion

When you travel abroad you change euro into another currency. The exchange rate tells you how much one euro is worth.

If €1 = US$1.08, then to change euro into dollars you **multiply** by $1.08$. To change dollars back into euro you **divide** by $1.08$.

Speed, distance and time

Speed tells you how far you go in one unit of time. The three quantities are linked by:

Cover the quantity you want. If you cover distance you see speed × time. If you cover speed you see distance over time.

The units must match. If the speed is in km/h, the time must be in hours. Change minutes to hours by dividing by 60: 45 minutes is $0.75$ hours, and $2\text{ h } 15\text{ min} = 2.25$ hours. To go back, multiply the decimal part by 60: $2.4\text{ h} = 2\text{ h } 24\text{ min}$ because $0.4 \times 60 = 24$.

Average speed

Average speed is the total distance divided by the total time. It is not the average of the speeds.

Back to Cian: his bus travels 20 km in 40 minutes, which is $\tfrac{2}{3}$ h, so its speed is $20 \div \tfrac{2}{3} = 30$ km/h.

Summary

• Ratio: add the parts, divide the total by that number to get one part, then multiply. Simplify by dividing by the HCF.

• VAT and percentages: use multipliers ($1.23$ to add $23\%$). To undo, divide by the multiplier.

• Compound interest: $F = P(1+i)^t$. Divide consecutive values to find the multiplier, and round only at the end.

• Pay and tax: gross pay, then tax at $20\%$ and $40\%$, subtract tax credits from the tax, then take off USC and PRSI to get net pay.

• Currency: multiply by the rate to go from euro, divide to go back to euro.

• Speed: $S = D \div T$, with matching units. Average speed is total distance over total time.