Financial mathematics deals with money that increases or decreases over time. In every calculation, identify the principal amount, interest rate per payment or compounding period, and number of periods. Convert percentages to decimals and ensure that the interest period agrees with the time period.<\/p>\n\n\n\n
The examination-style methods and values below follow the financial mathematics solutions in the supplied marking guidance.<\/p>\n\n\n\n
Simple interest is calculated only on the original principal. The interest earned during each period is therefore constant.<\/p>\n\n\n\n
The formula is:<\/p>\n\n\n\n
Here:<\/p>\n\n\n\n
The simple interest earned is:<\/p>\n\n\n\n
Example<\/strong><\/p>\n\n\n\nCalculate the accumulated amount when R12 000 is invested for 4 years at 8,5% simple interest per annum.<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>n<\/mi>)<\/mo><\/mrow>A=P(1+in)<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>12<\/mn><\/mspace>000<\/mn>(<\/mo>1<\/mn>+<\/mo>0,085<\/mn>(<\/mo>4<\/mn>)<\/mo>)<\/mo><\/mrow><\/mrow>A=12\\,000\\left(1+0{,}085(4)\\right)<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>12<\/mn><\/mspace>000<\/mn>(<\/mo>1,34<\/mn>)<\/mo><\/mrow>A=12\\,000(1{,}34)<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>R<\/mtext>16<\/mn><\/mspace>080<\/mn><\/mrow>A=\\text{R}16\\,080<\/annotation><\/semantics><\/math><\/div>\n\n\n\nThe interest earned is:<\/p>\n\n\n\nI<\/mi>=<\/mo>A<\/mi>\u2212<\/mo>P<\/mi><\/mrow>I=A-P<\/annotation><\/semantics><\/math><\/div>\n\n\n\nI<\/mi>=<\/mo>16<\/mn><\/mspace>080<\/mn>\u2212<\/mo>12<\/mn><\/mspace>000<\/mn><\/mrow>I=16\\,080-12\\,000<\/annotation><\/semantics><\/math><\/div>\n\n\n\nI<\/mi>=<\/mo>R<\/mtext>4<\/mn><\/mspace>080<\/mn><\/mrow>I=\\text{R}4\\,080<\/annotation><\/semantics><\/math><\/div>\n\n\n\nCommon Mistake<\/strong><\/p>\n\n\n\nDo not use simple interest when the question states that interest is compounded.<\/p>\n\n\n\n<\/a><\/figure>\n\n\n\nCompound interest<\/h2>\n\n\n\nCompound interest is calculated on the principal and on interest already earned. This produces exponential growth.<\/p>\n\n\n\nThe formula is:<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR40 000 is invested at 7,8% per annum, compounded annually, for 5 years. Calculate the accumulated amount.<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1<\/mn>+<\/mo>0,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1+0{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>R<\/mtext>58<\/mn><\/mspace>230,94<\/mn><\/mrow>A=\\text{R}58\\,230{,}94<\/annotation><\/semantics><\/math><\/div>\n\n\n\nIf compounding occurs several times per year, adjust both the interest rate and the number of periods.<\/p>\n\n\n\ni<\/mi>=<\/mo>i<\/mi>nom<\/mtext><\/msub>m<\/mi><\/mfrac><\/mrow>i=\\frac{i_{\\text{nom}}}{m}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>m<\/mi>t<\/mi><\/mrow>n=mt<\/annotation><\/semantics><\/math><\/div>\n\n\n\nHere, m is the number of compounding periods per year and t is the number of years.<\/p>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR25 000 is invested for 3 years at 9% per annum, compounded monthly.<\/p>\n\n\n\ni<\/mi>=<\/mo>0,09<\/mn>12<\/mn><\/mfrac><\/mrow>i=\\frac{0{,}09}{12}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>12<\/mn>(<\/mo>3<\/mn>)<\/mo>=<\/mo>36<\/mn><\/mrow>n=12(3)=36<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>25<\/mn><\/mspace>
Calculate the accumulated amount when R12 000 is invested for 4 years at 8,5% simple interest per annum.<\/p>\n\n\n\n
The interest earned is:<\/p>\n\n\n\n
Common Mistake<\/strong><\/p>\n\n\n\nDo not use simple interest when the question states that interest is compounded.<\/p>\n\n\n\n<\/a><\/figure>\n\n\n\nCompound interest<\/h2>\n\n\n\nCompound interest is calculated on the principal and on interest already earned. This produces exponential growth.<\/p>\n\n\n\nThe formula is:<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR40 000 is invested at 7,8% per annum, compounded annually, for 5 years. Calculate the accumulated amount.<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1<\/mn>+<\/mo>0,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1+0{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>R<\/mtext>58<\/mn><\/mspace>230,94<\/mn><\/mrow>A=\\text{R}58\\,230{,}94<\/annotation><\/semantics><\/math><\/div>\n\n\n\nIf compounding occurs several times per year, adjust both the interest rate and the number of periods.<\/p>\n\n\n\ni<\/mi>=<\/mo>i<\/mi>nom<\/mtext><\/msub>m<\/mi><\/mfrac><\/mrow>i=\\frac{i_{\\text{nom}}}{m}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>m<\/mi>t<\/mi><\/mrow>n=mt<\/annotation><\/semantics><\/math><\/div>\n\n\n\nHere, m is the number of compounding periods per year and t is the number of years.<\/p>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR25 000 is invested for 3 years at 9% per annum, compounded monthly.<\/p>\n\n\n\ni<\/mi>=<\/mo>0,09<\/mn>12<\/mn><\/mfrac><\/mrow>i=\\frac{0{,}09}{12}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>12<\/mn>(<\/mo>3<\/mn>)<\/mo>=<\/mo>36<\/mn><\/mrow>n=12(3)=36<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>25<\/mn><\/mspace>
Do not use simple interest when the question states that interest is compounded.<\/p>\n\n\n\n<\/a><\/figure>\n\n\n\nCompound interest<\/h2>\n\n\n\nCompound interest is calculated on the principal and on interest already earned. This produces exponential growth.<\/p>\n\n\n\nThe formula is:<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR40 000 is invested at 7,8% per annum, compounded annually, for 5 years. Calculate the accumulated amount.<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1<\/mn>+<\/mo>0,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1+0{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>R<\/mtext>58<\/mn><\/mspace>230,94<\/mn><\/mrow>A=\\text{R}58\\,230{,}94<\/annotation><\/semantics><\/math><\/div>\n\n\n\nIf compounding occurs several times per year, adjust both the interest rate and the number of periods.<\/p>\n\n\n\ni<\/mi>=<\/mo>i<\/mi>nom<\/mtext><\/msub>m<\/mi><\/mfrac><\/mrow>i=\\frac{i_{\\text{nom}}}{m}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>m<\/mi>t<\/mi><\/mrow>n=mt<\/annotation><\/semantics><\/math><\/div>\n\n\n\nHere, m is the number of compounding periods per year and t is the number of years.<\/p>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR25 000 is invested for 3 years at 9% per annum, compounded monthly.<\/p>\n\n\n\ni<\/mi>=<\/mo>0,09<\/mn>12<\/mn><\/mfrac><\/mrow>i=\\frac{0{,}09}{12}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>12<\/mn>(<\/mo>3<\/mn>)<\/mo>=<\/mo>36<\/mn><\/mrow>n=12(3)=36<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>25<\/mn><\/mspace>
Compound interest is calculated on the principal and on interest already earned. This produces exponential growth.<\/p>\n\n\n\n
Example<\/strong><\/p>\n\n\n\nR40 000 is invested at 7,8% per annum, compounded annually, for 5 years. Calculate the accumulated amount.<\/p>\n\n\n\nA<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>i<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A=P(1+i)^n<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1<\/mn>+<\/mo>0,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1+0{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>40<\/mn><\/mspace>000<\/mn>(<\/mo>1,078<\/mn>)<\/mo>5<\/mn><\/msup><\/mrow>A=40\\,000(1{,}078)^5<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>R<\/mtext>58<\/mn><\/mspace>230,94<\/mn><\/mrow>A=\\text{R}58\\,230{,}94<\/annotation><\/semantics><\/math><\/div>\n\n\n\nIf compounding occurs several times per year, adjust both the interest rate and the number of periods.<\/p>\n\n\n\ni<\/mi>=<\/mo>i<\/mi>nom<\/mtext><\/msub>m<\/mi><\/mfrac><\/mrow>i=\\frac{i_{\\text{nom}}}{m}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>m<\/mi>t<\/mi><\/mrow>n=mt<\/annotation><\/semantics><\/math><\/div>\n\n\n\nHere, m is the number of compounding periods per year and t is the number of years.<\/p>\n\n\n\nExample<\/strong><\/p>\n\n\n\nR25 000 is invested for 3 years at 9% per annum, compounded monthly.<\/p>\n\n\n\ni<\/mi>=<\/mo>0,09<\/mn>12<\/mn><\/mfrac><\/mrow>i=\\frac{0{,}09}{12}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>12<\/mn>(<\/mo>3<\/mn>)<\/mo>=<\/mo>36<\/mn><\/mrow>n=12(3)=36<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>25<\/mn><\/mspace>
R40 000 is invested at 7,8% per annum, compounded annually, for 5 years. Calculate the accumulated amount.<\/p>\n\n\n\n
If compounding occurs several times per year, adjust both the interest rate and the number of periods.<\/p>\n\n\n\n
Here, m is the number of compounding periods per year and t is the number of years.<\/p>\n\n\n\n
Example<\/strong><\/p>\n\n\n\nR25 000 is invested for 3 years at 9% per annum, compounded monthly.<\/p>\n\n\n\ni<\/mi>=<\/mo>0,09<\/mn>12<\/mn><\/mfrac><\/mrow>i=\\frac{0{,}09}{12}<\/annotation><\/semantics><\/math><\/div>\n\n\n\nn<\/mi>=<\/mo>12<\/mn>(<\/mo>3<\/mn>)<\/mo>=<\/mo>36<\/mn><\/mrow>n=12(3)=36<\/annotation><\/semantics><\/math><\/div>\n\n\n\nA<\/mi>=<\/mo>25<\/mn><\/mspace>
R25 000 is invested for 3 years at 9% per annum, compounded monthly.<\/p>\n\n\n\n