Unit two | general mathematics Financial arithmetic Unit two | general mathematics

Percentage change Calculating the Percentage Change The percentage change is found by taking the change that has occurred and expressing it as a percentage of the starting value. Calculating the Percentage Change % 𝑐ℎ𝑎𝑛𝑔𝑒= 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑃𝑟𝑖𝑐𝑒 ×100

Percentage change % 𝑐ℎ𝑎𝑛𝑔𝑒= 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑃𝑟𝑖𝑐𝑒 ×100 eg1. A pair of shoes is marked down from $125 to $80. Find the percentage change of the shoes. % 𝑐ℎ𝑎𝑛𝑔𝑒= 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑃𝑟𝑖𝑐𝑒 ×100 = 45 125 ×100 =36 %

Percentage change % 𝑐ℎ𝑎𝑛𝑔𝑒= 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑃𝑟𝑖𝑐𝑒 ×100 eg2. Petrol increases in price from $1.20 per litre to $1.35 per litre. Find the percentage change in the cost of petrol. % 𝑐ℎ𝑎𝑛𝑔𝑒= 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑃𝑟𝑖𝑐𝑒 ×100 = 15 120 ×100 =12.5 %

Unit two | general mathematics Now do Ex3.2 q1ac, 2, 9, 14, 15 Unit two | general mathematics

Finding the new cost of a marked up / down item When an item is marked down (reduced in price) by a percentage, r, we can find the new price of the item using: When an item is marked up (increases in price) by a percentage, r, we can find the new price of the item using: 𝑛𝑒𝑤 𝑝𝑟𝑖𝑐𝑒=𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 × (100−𝑟) 100 𝑛𝑒𝑤 𝑝𝑟𝑖𝑐𝑒=𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 × (100+𝑟) 100

marked down (reduced) marked up (increased) eg marked down (reduced) marked up (increased) eg. Reduce $240 by 25% 𝑛𝑒𝑤 𝑝𝑟𝑖𝑐𝑒=𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 × (100−𝑟) 100 =240× (100−25) 100 =240× 75 100 = $180 𝑛𝑒𝑤 𝑝𝑟𝑖𝑐𝑒=𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 × (100−𝑟) 100 𝑛𝑒𝑤 𝑝𝑟𝑖𝑐𝑒=𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 × (100+𝑟) 100 eg. Increase $130 by 15% 𝑛𝑒𝑤 𝑝𝑟𝑖𝑐𝑒=𝑜𝑙𝑑 𝑝𝑟𝑖𝑐𝑒 × (100+𝑟) 100 =130× (100+15) 100 =130× 115 100 = $149.50

Unit two | general mathematics Now do Ex3.2 Q3ac, 4ac, 5, 7, 13 Ex3.3 Q7 Unit two | general mathematics

Price changes using a spreadsheet

Unit two | general mathematics Now do Ex3.3 Q22 Unit two | general mathematics

Shares and dividends When you buy shares in a company, you effectively own a portion of the company. If the company isn’t going well, the price you paid for your shares decreases in value. Likewise, when it’s successful, the value of your share increases in value. Any profit made is distributed to shareholders and paid as a dividend. 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒= 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑟𝑒𝑠 eg. A company with 500 000 shares makes an annual profit of $340 000, Calculate the dividend payable per share. 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 = 340 000 500 000 =0.68 =68 cents per share

Shares and dividends 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒= 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑟𝑒𝑠 eg2. A company declares a total dividend of $480 000 and a dividend per share of 60 cents. How many shares are in the company? 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 = 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑟𝑒𝑠 Number of Shares = 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 = 480 000 0.60 =800 000

Percentage dividend Shares in different companies vary in price drastically from cents to hundreds of $$. When deciding to buy shares, investors often look at the company’s Percentage Dividend. This shows the investor what proportion of their investment comes back to them as a dividend. 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑= 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝑆ℎ𝑎𝑟𝑒 𝑝𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 ×100 eg. Calculate the percentage dividend on a share that cost $1.25 and paid a dividend of 8 cents per share. 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 = 0.08 1.25 ×100 =6.4 %

Unit two | general mathematics Now do Ex3.3 Q1,2,3 Unit two | general mathematics

Price-to-earnings ratio The price-to-earnings ratio (P/E Ratio) is another way of comparing shares, by looking at the current share price and the annual dividend. The P/E Ratio gives us an indication of how much shares cost per dollar of profit. 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜 = 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 eg. Calculate the price-to-earnings ratio for a company with a share price of $2.20 and gives a dividend of 10 cents. 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜= 2.20 0.10 =22

Price-to-earnings ratio 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜 = 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 eg2. Calculate the price of a share if the price-to-earnings ratio is 15 and pays a dividend of 20 cents. 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜= 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒=𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜 ×𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 =15×0.20 =$3.00

Price-to-earnings ratio 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜 = 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 eg3. Calculate the dividend paid per share if the price-to-earnings ratio is 36 and the share price is $1.44 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜= 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒= 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜 = 1.44 36 =0.04 =4 𝑐𝑒𝑛𝑡𝑠

What is the dividend per share payable to shareholders? 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜 = 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒= 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑟𝑒𝑠 eg4. A company that has 800 000 shares declares an annual gross profit of $8 400 260. They pay 20% of this in tax, and reinvests 30% of the remaining profit (net profit). What is the dividend per share payable to shareholders? What is the Price-to-earnings ratio if the current share price is $38.90 ? 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒= 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒𝑠 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 → 20% paid in tax, 80% remaining after tax =8 400 260 × 0.80 =$6,720,208 30% of total after tax reinvested, leaving 70% dividend =6 720 208 ×0.70 =$4,407,145.60 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒= 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒𝑠 = 4 407 145.60 800 000 =$5.88 𝑃/𝐸 𝑅𝑎𝑡𝑖𝑜= 𝑆ℎ𝑎𝑟𝑒 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 = 38.90 5.88 =6.62

Unit two | general mathematics Now do Ex3.3 Q5,6,11ac,13 Unit two | general mathematics

Calculating cost with & without G.S.T. GST is a 10% Goods and Services Tax imposed on some products and services. We can use the following chart to calculate the price of goods with/without GST eg1. The price of a bottle of Juice is $4.20 before GST is added. Find the cost of the item if GST was added to this price. 𝑃𝑟𝑖𝑐𝑒 𝑤𝑖𝑡ℎ 𝐺𝑆𝑇= 4.20 ×1.1 =$4.62

Calculating cost with & without G.S.T. eg2. The price of dress is $ 89.00 before GST is added. Find the cost of the item if GST was added to this price. 𝑃𝑟𝑖𝑐𝑒 𝑤𝑖𝑡ℎ 𝐺𝑆𝑇=89.00 ×1.1 =$97.90

Calculating cost with & without G.S.T. eg3. The price of corn flakes is $4.90 including GST. Find the price of the item before GST was added. 𝑃𝑟𝑖𝑐𝑒 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝐺𝑆𝑇= 4.90 1.1 =$4.45

Calculating the G.S.T. To find the amount of GST on an item, decide whether the price already includes GST and choose the formula which applies. eg1. Calculate the GST on an item, if the price is $79.00 with GST included. eg2. Calculate the GST on an item, if the price is $53.40 before GST 𝐺𝑆𝑇=53.40 ×0.1 =$5.34 × 0.1 ÷11 𝐺𝑆𝑇= 79.00 11 =$7.18

Unit two | general mathematics Now do Ex3.3 Q17, 9, 10, 12, 20 Unit two | general mathematics

Simple interest 𝐼= 𝑃 × 𝑟 × 𝑡 100 When you borrow money, you pay for the privilege of lending and spending someone else’s money by paying them interest. In the same way, when you invest money you can get interest paid on your money. Simple Interest is one type of interest, calculated using: 𝑆𝑖𝑚𝑝𝑙𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡= 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 ×𝑅𝑎𝑡𝑒 ×𝑇𝑖𝑚𝑒 100 𝐼= 𝑃 × 𝑟 × 𝑡 100 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 −𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦 𝑏𝑜𝑟𝑟𝑜𝑤𝑒𝑑 𝑜𝑟 𝑑𝑒𝑝𝑜𝑠𝑖𝑡𝑒𝑑 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑡=𝑡𝑖𝑚𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑚𝑜𝑛𝑒𝑦 𝑖𝑠 𝑏𝑜𝑟𝑟𝑜𝑤𝑒𝑑/𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑓𝑜𝑟 (𝑦𝑒𝑎𝑟𝑠)

𝑆𝑖𝑚𝑝𝑙𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡= 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 ×𝑅𝑎𝑡𝑒 ×𝑇𝑖𝑚𝑒 100 𝐼= 𝑃 × 𝑟 × 𝑡 100 Eg1. Calculate the simple interest payable on a loan of $40 000, borrowed at a rate of 12% over 5 years. 𝐼= 𝑃 × 𝑟 × 𝑡 100 = 40 000 × 12 × 5 100 =$24 000 How much will the borrower owe to the bank in total? 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑦𝑎𝑏𝑙𝑒=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙+𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 =40 000+ 24 000 = $64 000

𝑆𝑖𝑚𝑝𝑙𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡= 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 ×𝑅𝑎𝑡𝑒 ×𝑇𝑖𝑚𝑒 100 𝐼= 𝑃 × 𝑟 × 𝑡 100 Eg2. A term deposit of $30,000 attracts 4.5% interest over 6 years. How much interest is made? 𝐼= 𝑃 × 𝑟 × 𝑡 100 = 30 000 × 4.5 × 6 100 =$8 100

𝑆𝑖𝑚𝑝𝑙𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡= 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 ×𝑅𝑎𝑡𝑒 ×𝑇𝑖𝑚𝑒 100 𝐼= 𝑃 × 𝑟 × 𝑡 100 Eg3. A $13,500 loan attracts an interest rate of 16% over 90 months. How much interest is paid on this loan? Convert 90 months into years: 90 𝑚𝑜𝑛𝑡ℎ𝑠 12 𝑚𝑜𝑛𝑡ℎ𝑠 =7.5 𝑦𝑒𝑎𝑟𝑠 𝐼= 𝑃 × 𝑟 × 𝑡 100 = 13 500 × 16 × 7.5 100 =$16 200 What is the total amount payable on the loan? Total = Principal + Interest = 13 500 + 16 200 = $29 700

𝐼= 𝑃 × 𝑟 × 𝑡 100 Eg4. A $10 000 loan attracts an interest rate of 11% over 3 years and 3 months a) How much interest is paid on this loan? Convert 3 years and 3 months into years: 3 𝑦𝑒𝑎𝑟𝑠+ 3 𝑚𝑜𝑛𝑡ℎ𝑠 12 𝑚𝑜𝑛𝑡ℎ𝑠 =3.25 𝑦𝑒𝑎𝑟𝑠 𝐼= 𝑃 × 𝑟 × 𝑡 100 = 10 000 × 11 × 3.25 100 =$3 575 b) What is the total amount payable on the loan? Total = Principal + Interest = 10 000 + 3 575 = $13 575 c) If making equal monthly repayments, how much would I be charged each month? 13 575 3 ×12 +3 = 13 575 39 =$348.08

Unit two | general mathematics Now do Ex3.4 Q1, 5, 6 Unit two | general mathematics

Simple interest We can rearrange the formula to solve other unknowns:

𝑃= 100 × 𝐼 𝑟 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 𝑟= 100 × 𝐼 𝑃 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 Solving the Principal ($) Solving the Rate (%) Solving the Time (years) 𝑃= 100 × 𝐼 𝑟 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 𝑟= 100 × 𝐼 𝑃 × 𝑡 eg1. Find the time period of a $2200 loan which produces interest of $440 at 8% interest per annum. 𝑡= 100 × 𝐼 𝑟 × 𝑃 = 100 × 440 8 ×2200 = 44000 17600 =2.5 𝑦𝑒𝑎𝑟𝑠 𝑡=? 𝐼=$440 𝑟=8% 𝑃=$2200

𝑃= 100 × 𝐼 𝑟 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 𝑟= 100 × 𝐼 𝑃 × 𝑡 𝑟= 100 × 𝐼 𝑃 × 𝑡 Solving the Principal ($) Solving the Rate (%) Solving the Time (years) 𝑃= 100 × 𝐼 𝑟 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 𝑟= 100 × 𝐼 𝑃 × 𝑡 eg2. Find the rate of an $8000 loan which produces interest of $1750 over 5 years 𝑟= 100 × 𝐼 𝑃 × 𝑡 = 100 × 1750 8000 × 5 = 175000 40000 =4.375 % 𝑟=? 𝐼=$1750 𝑡=5 𝑃=$8000

𝑃= 100 × 𝐼 𝑟 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 𝑟= 100 × 𝐼 𝑃 × 𝑡 𝑃= 100 × 𝐼 𝑟 × 𝑡 Solving the Principal ($) Solving the Rate (%) Solving the Time (years) 𝑃= 100 × 𝐼 𝑟 × 𝑡 𝑡= 100 × 𝐼 𝑟 × 𝑃 𝑟= 100 × 𝐼 𝑃 × 𝑡 eg3. Find the principal of a loan that attracts $927.50 interest over 3.5 years at a rate of 13.25% per annum 𝑃= 100 × 𝐼 𝑟 × 𝑡 = 100 × 927.50 13.25 × 3.5 = 92750 46.375 = $2000 𝑟=? 𝐼=$1750 𝑡=5 𝑃=$8000

Unit two | general mathematics Now do Ex3.4 Q3, 4, 16, 18, 22, 13 Unit two | general mathematics

CASh flow - Simple interest non-annual interest calculations Interest rates are usually advertised at a per annum rate, however in reality interest is usually accrued more frequently than this – Quarterly, Monthly, Weekly, Daily etc. eg1. How much interest is paid on a monthly balance of $900 with a simple interest rate of 11% per annum? 𝐼= 𝑃𝑟𝑡 100 𝐼= 900×11× 1 12 100 𝐼=$8.25 P=900 r=11 t= 1 12

CASh flow - Simple interest non-annual interest calculations eg2. How much interest is paid on a weekly balance of $1400 with a simple interest rate of 8.2% per annum? 𝐼= 𝑃𝑟𝑡 100 𝐼= 1400 × 8.2 × 1 52 100 𝐼=$2.21 P=1400 r=8.2 t= 1 52

CASh flow - Simple interest non-annual interest calculations eg3. How much interest is paid on a quarterly balance of $3200 with a simple interest rate of 6.8% per annum? 𝐼= 𝑃𝑟𝑡 100 𝐼= 3200 × 6.8 × 1 4 100 𝐼=$54.40 P=3200 r=6.8 t= 1 4

CASh flow - Simple interest minimum balance calculations When applying interest to an account, it is common practice to apply interest to the minimum balance during a set time period. In investment accounts, the bank finds the lowest amount of money that you have in the account during that time and applies interest to that amount. eg1. How much interest is paid on this account if interest paid monthly at 3% per annum and the opening balance is $1500 ? $1500 − $150 = $1350 $1350 + $220 = $1570 Minimum Balance $1570 − $500 = $1070 $1350 + $120 = $1470

CASh flow - Simple interest minimum balance calculations Eg1 (continued). How much interest is paid on this account if interest paid monthly at 3% per annum and the opening balance is $1500 ? $1500 − $150 = $1350 $1350 + $220 = $1570 Minimum Balance $1570 − $500 = $1070 $1350 + $120 = $1470 𝐼= 𝑃𝑟𝑡 100 𝐼= 1070 × 3 × 1 12 100 𝐼=$2.68 P=1070 r=3 t= 1 12

Unit two | general mathematics Now do Ex3.4 Q7ab,8,9,14 Unit two | general mathematics

compound interest With simple interest, the interest paid is constant and spread out over a loan period. Compound Interest is different as interest is calculated and added periodically throughout the loan. This means that the dollar amount of interest paid changes in each period based on the balance of the loan for that period.

compound interest – paid annually Calculating Compound Interest with interest paid annually: 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 −𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦 𝑏𝑜𝑟𝑟𝑜𝑤𝑒𝑑 𝑜𝑟 𝑑𝑒𝑝𝑜𝑠𝑖𝑡𝑒𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑙𝑦 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 =5,000 (1+ 4 100 ) 1 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 COMPOUND INTEREST 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 eg1. If I invest $5000 in a compound deposit at 4% interest per annum paid annually: a) How much do I have after one year? 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 After FIRST year 𝑃=$5,000 𝑟=4% 𝑛=1 =5,000 (1+ 4 100 ) 1 =5,000 (1.04 ) 1 =$5,200

𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 =5,000 (1+ 4 100 ) 5 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 COMPOUND INTEREST 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 eg1. If I invest $5000 in a compound deposit at 4% interest per annum paid annually: b) How much do I have after 5 years? 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 After 5 years 𝑃=$5,000 𝑟=4% 𝑛=5 =5,000 (1+ 4 100 ) 5 =5,000 (1.04 ) 5 =$6,083.26 How much Interest has been earned? 𝐼=𝐴−𝑃=6083.26−5000=$1083.26

𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 =8,000 (1+ 5 100 ) 2 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 COMPOUND INTEREST 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 eg2. If I invest $8000 in a compound deposit at 5% interest per annum paid annually: a) How much do I have after 2 years? 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 After 2 years 𝑃=$8,000 𝑟=5% 𝑛=2 =8,000 (1+ 5 100 ) 2 =8,000 (1.05 ) 2 =$8,820

𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 =8,000 (1+ 5 100 ) 5 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 COMPOUND INTEREST 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 eg2. If I invest $8000 in a compound deposit at 5% interest per annum paid annually: b) How much do I have after 5 years? 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 After 5 years 𝑃=$8,000 𝑟=5% 𝑛=5 =8,000 (1+ 5 100 ) 5 =8,000 (1.05 ) 5 =$10,210.25

compound interest 𝑃= 𝐴 (1+ 𝑟 100 ) 𝑛 𝑟=100 (( 𝐴 𝑃 ) 1 𝑛 −1) 𝑃= 𝐴 (1+ 𝑟 100 ) 𝑛 Finding the Principal: 𝑟=100 (( 𝐴 𝑃 ) 1 𝑛 −1) Finding the Rate: 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 −𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦 𝑏𝑜𝑟𝑟𝑜𝑤𝑒𝑑 𝑜𝑟 𝑑𝑒𝑝𝑜𝑠𝑖𝑡𝑒𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑙𝑦 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

𝑃= 𝐴 (1+ 𝑟 100 ) 𝑛 𝑟=100 (( 𝐴 𝑃 ) 1 𝑛 −1) eg1. A car loan with a rate of 8.9% p.a. over 5 years, has a final value of $62,423. What was the purchase price of the car? b) Another loan option attracted $18,000 interest over 4 years, compounding annually. What rate was this option? 𝑃= 𝐴 (1+ 𝑟 100 ) 𝑛 = 62423 (1+ 8.9 100 ) 5 =$40,757.28 𝑟=8.9 𝐴=62423 𝑛=5 𝑟= ? 𝑃=40757.28 𝐴=40757.28+18000 =58757.28 𝑛=4 𝑟=100( 58757.28 40757.28 ) 1 4 −1 = 9.58%

Unit two | general mathematics Now do Ex3.5 Q2,4,1,3,5,6 Unit two | general mathematics

compound interest – non-annual Calculating Compound Interest with interest paid non-annually: 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 𝐴=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑝𝑙𝑢𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 −𝑡ℎ𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑚𝑜𝑛𝑒𝑦 𝑏𝑜𝑟𝑟𝑜𝑤𝑒𝑑 𝑜𝑟 𝑑𝑒𝑝𝑜𝑠𝑖𝑡𝑒𝑑 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑙𝑦 𝑟=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑎𝑛𝑛𝑢𝑚 (%) 𝑛=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

compound interest – non-annual 𝐴=𝑃 (1+ 𝑟 400 ) 𝑛 Compounding Quarterly: Compounding Monthly: Compounding Weekly: Compounding Fortnightly: Compounding Daily: 𝐴=𝑃 (1+ 𝑟 1200 ) 𝑛 𝐴=𝑃 (1+ 𝑟 5200 ) 𝑛 𝐴=𝑃 (1+ 𝑟 2600 ) 𝑛 𝐴=𝑃 (1+ 𝑟 36500 ) 𝑛

𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 Eg1. Calculate the final balance for a $3000 investment at 4% pa for 5 years, compounding weekly 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 =3000 1+ 4 5200 260 =$3663.93 A = ? P = 3000 r = 4 n = (5 x 52) = 260 Compoundings /yr = 52 What if it was compounding daily? 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 =3000 1+ 4 36500 1825 =$3664.17

𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 Eg2. Calculate the Interest on a $20000 loan at 12% per annum for 4 years, compounding a) Monthly b) Quarterly c) Fortnightly MONTHLY 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 =20000 1+ 12 1200 48 =$32 244.52 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝐴 − 𝑃 =32 244.52 −20 000 =$12 244.52 A = ? P = 20000 r = 12 n = (4 x 12) = 48 Compoundings /yr = 12

𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 Eg2. Calculate the Interest on a $20000 loan at 12% per annum for 4 years, compounding a) Monthly b) Quarterly c) Fortnightly QUARTERLY 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 =20000 1+ 12 400 16 =$32 094.13 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝐴 − 𝑃 =32 094.13 −20 000 =$12 094.13 A = ? P = 20000 r = 12 n = (4 x 4) = 16 Compoundings /yr = 4

𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 Eg2. Calculate the Interest on a $20000 loan at 12% per annum for 4 years, compounding a) Monthly b) Quarterly c) Fortnightly FORTNIGHTLY 𝐴=𝑃 (1+ 𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑝𝑜𝑢𝑑𝑖𝑛𝑔𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟×100 ) 𝑛 =20000 1+ 12 2600 104 =$32 285.82 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝐴 − 𝑃 =32 285.82 −20 000 =$12 285.82 A = ? P = 20000 r = 12 n = (4 x 26) = 104 Compoundings /yr = 26

Unit two | general mathematics Now do Ex3.5 Q8,11,12,16 Unit two | general mathematics

Applications of compound interest – inflation Inflation is a term used to describe a general increase in prices over time. In reality, when analyzing profits and losses over a period of time, we must take inflation into consideration. Inflation can be measured by the inflation rate, which is an annual percentage change of the CPI (Consumer Price Index) For example – A house costs $350 000 and sells for $390 000 5 years later. Considering prices usually increase over time, has a profit really been made? Or has the price just increased as per the rate of inflation for that time period?

Applications of compound interest – inflation Eg. An investment property is purchased for $250 000 and is sold 4 years later for $290 000. If the annual average inflation is 1.7% per annum, has this been a profitable investment? Inflation is an application of compound interest – find the value of $250,000, 4 years on…… 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 =250 000 1+ 1.7 100 4 =250 000 (1.017) 4 =$267 438.43 Inflated amount = $267 438.43. Sell Price = $290 000 They have made a real profit of $22 561.57 𝑷=𝟐𝟓𝟎 𝟎𝟎𝟎 𝒓=𝟏.𝟕 𝒏=𝟒 𝑨= ?

Applications of compound interest – inflation Eg2. An business is purchased for $120 000 and is sold 3 years later for $125000. If the annual average inflation is 2.1% per annum, has a real profit been made? Inflation is an application of compound interest – find the value of $120000, 3 years on…… 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 =120 000 1+ 2.1 100 3 =120 000 (1.021) 3 =$127 719.87 Inflated amount = $127 719.87. Sell Price = $125 000 They have not made a real profit. 𝑷=𝟏𝟐𝟎 𝟎𝟎𝟎 𝒓=𝟐.𝟏 𝒏=𝟑 𝑨= ?

Applications of compound interest – inflation Eg3. A person invests $2000 wanting to purchase a $2500 item in 3 years. They reach this goal by investing in an account paying compound interest quarterly. What rate is the investment paying? The item increases with inflation at a rate of 1.9% pa. How much does the item actually cost at the end of the 3 years? How much should they have invested to buy the item at the inflated price? a) 𝑷=𝟐𝟎𝟎𝟎 𝒓= ? 𝒏=𝟑 ×𝟒 𝑨=𝟐𝟓𝟎𝟎

Applications of compound interest – inflation Eg3. A person invests $2000 wanting to purchase a $2500 item in 3 years. b) The item increases with inflation at a rate of 1.9% pa. How much does the item actually cost at the end of the 3 years? 𝐴=𝑃 (1+ 𝑟 100 ) 𝑛 = 2500 1+ 1.9 100 3 = 2500 1.019 3 =$2645.22 𝑷=𝟐𝟓𝟎𝟎 𝒓=𝟏.𝟗 𝒏=𝟑 𝑨= ? c) How much should they have invested to buy the item at the inflated price?

Applications of compound interest – inflation Eg3. A person invests $2000 wanting to purchase a $2500 item in 3 years. c) How much should they have invested to buy the item at the inflated price? 𝐴=𝑃 (1+ 𝑟 400 ) 𝑛 𝑃= 𝐴 (1+ 𝑟 400 ) 𝑛 𝑷=? 𝒓=𝟕.𝟓𝟏 𝒏=𝟑 ×𝟒 𝑨=𝟐𝟔𝟒𝟓.𝟐𝟐 𝑃= 2645.22 (1+ 7.51 400 ) 12 =$2116.03

Unit two | general mathematics Now do Ex3.5 Q9,10,15,17 Unit two | general mathematics