Numbers, Fractions, Decimals, and Percents
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These are the solutions to the CSEC past questions on Numbers, Fractions, Decimals, and Percents.
The TI-84 Plus CE shall be used for applicable questions.
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For calculations involving scientific notation as applicable, change the mode to SCI.
Percent Applications
$ \underline{\text{Change and Percent of Change}} \\[3ex] (1.)\:\: change = new - initial \\[5ex] (2.)\:\: \%\;\;of\;\;change = \dfrac{change}{initial} * 100 \\[5ex] (3.)\;\; \text{If A is p% more than B, then A = (100 + p)% of B} \\[3ex] (4.)\;\; \text{If A is p% less than B, then A = (100 – p)% of B} \\[5ex] \underline{\text{Percent:Proportion}} \\[3ex] (5.)\;\; \dfrac{is}{of} = \dfrac{\%}{100} \\[5ex] (6.)\;\;\underline{\text{Percent:Equation}} \\[3ex] is \rightarrow equal\;\;to \\[3ex] of\;\; \rightarrow multiply \\[3ex] what \;\; \rightarrow variable \\[5ex] \underline{\text{Wholesale and Retail}} \\[3ex] (7.)\:\: \text{Sale Price } = \text{Initial Price } - \text{Discount} \\[5ex] (8.)\:\: \%\;Discount = \dfrac{Discount}{\text{Initial Price}} * 100 \\[5ex] (9.)\;\; \text{Profit } = \text{Selling Price } - \text{Cost Price} \\[3ex] (10.)\;\; \%\;Profit = \dfrac{Profit}{\text{Cost Price}} * 100 \\[5ex] (11.)\;\; \text{Loss } = \text{Cost Price } - \text{Selling Price} \\[3ex] (12.)\;\; \%\;Loss = \dfrac{Loss}{\text{Cost Price}} * 100 $
Formula Sheet: List of Formulae
(i.) $\dfrac{2}{3} \;\;of\;\; \left(\dfrac{1}{8} + \dfrac{5}{12} \div \dfrac{1}{9}\right)$, giving your answer in EXACT form.
(ii.) $314.2 - \dfrac{26082}{52164}$, giving your answer in standard form.
(b.) Jim packed several cases of fruit juice for sale.
Each case contained 24 boxes of juice in 3 different varieties: apple, orange, and pineapple in the ratio 2:5:1 respectively.
(i.) How many boxes of pineapple juice were packed in each case?
(ii.) The profit gained from selling ALL of the boxes of pineapple juice is $35.64.
Each box of pineapple juice was sold at $3.34.
(a.) Show that the cost price of a box of pineapple juice is $2.35
(b.) Calculate the percentage profit made on the sale of the boxes of pineapple juice.
$ (a.) (i.) \\[3ex] \dfrac{2}{3} \;\;of\;\; \left(\dfrac{1}{8} + \dfrac{5}{12} \div \dfrac{1}{9}\right) \\[5ex] \underline{PEMDAS} \\[3ex] \text{Parenthesis: Division and Addition} \\[3ex] Division: \\[3ex] \dfrac{5}{12} \div \dfrac{1}{9} \\[5ex] = \dfrac{5}{12} * \dfrac{9}{1} \\[5ex] = \dfrac{15}{4} \\[5ex] Addition: \\[3ex] \dfrac{1}{8} + \dfrac{15}{4} \\[5ex] = \dfrac{1}{8} + \dfrac{30}{8} \\[5ex] = \dfrac{31}{8} \\[5ex] of \implies Multiplication: \\[3ex] \dfrac{2}{3} \;\;of\;\; \dfrac{31}{8} \\[5ex] = \dfrac{2}{3} * \dfrac{31}{8} \\[5ex] = \dfrac{31}{12} \\[5ex] (ii.) \\[3ex] 314.2 - \dfrac{26082}{52164} \\[5ex] = 314.2 - 0.5 \\[3ex] = 313.7 \\[3ex] = 3.137 * 10^{2}\quad\dots\text{in standard form} \\[5ex] (b.)(i.) \\[3ex] \text{24 boxes containing 3 fruit varieties} \\[3ex] \text{Apple : Orange : Pineapple = 2 : 5 : 1} \\[3ex] \text{sum of ratios} = 2 + 5 + 1 = 8 \\[3ex] \text{number of boxes of pineapple juice} = \dfrac{1}{8} * 24 \\[5ex] = 3\;boxes \\[3ex] $ (ii.) (a.)
Selling Price, of each pineapple juice box = $3.34
Profit gained from selling all pineapple juice boxes = $35.64
But we are not given the number of pineapple juice boxes that were sold
To show that the cost price of a pineapple juice box = $2.35, we shall assume that 36 boxes of pineapple juice boxes were sold.
$ \underline{\text{36 boxes of pineapple juice}} \\[3ex] \text{Selling Price} = 3.34(36) = \$120.24 \\[3ex] Profit = \$35.64 \\[3ex] Profit = \text{Selling Price} - \text{Cost Price} \\[3ex] \text{Cost Price} = \text{Selling Price} - Profit \\[3ex] = 120.24 - 35.64 \\[3ex] = \$84.60 \\[3ex] \underline{\text{1 box of pineapple juice}} \\[3ex] C.P = \dfrac{84.60}{36} = \$2.35 \\[5ex] (b.) \\[3ex] \text{%Profit} = \dfrac{\text{Profit}}{\text{Cost Price}} * 100 \\[5ex] = \dfrac{35.64}{84.6} * 100 \\[5ex] = 42.12765957\% \\[3ex] \approx 42.1\%\quad\dots\text{to 3 significant figures} $
(1.)(a.)(i.)
(ii.)
$ (i.)\;\; \left(\dfrac{3}{2}\right)^2 - 1 + 6 \\[5ex] (ii.)\;\; \left(2.1 \times \dfrac{10}{7}\right) + 12 \div 1\dfrac{3}{5} \\[5ex] $ (b.) Maranda earns $8316 per month.
She spends $\dfrac{4}{7}$ of her earnings on utility bills, food and personal items, and saves the remainder.
(i.) What percentage of her monthly salary does she save?
(ii.) The portion of her earnings spent on utility bills, food and personal items is divided in the ratio 5 : 3 : 4 respectively.
Calculate the amount of money she spends on personal items.
(iii.) Maranda saves the same amount of money each month.
Show that her annual savings from her earnings is $42768.
(iv.) She invests the $42768 in her credit union which pays compound interest of 5% per annum.
What is the TOTAL interest earned after 2 years?
[The final amount, A, when principal, P, is invested compound interest at rate, r, for n number of years is given by $A = P\left(1 + \dfrac{r}{100}\right)^n$]
$ (a.)(i.) \\[3ex] \left(\dfrac{3}{2}\right)^2 - 1 + 6 \\[5ex] = \dfrac{9}{4} - \dfrac{4}{4} + \dfrac{24}{4} \\[5ex] = \dfrac{29}{4} \\[5ex] (ii.) \\[3ex] \left(2.1 \times \dfrac{10}{7}\right) + 12 \div 1\dfrac{3}{5} \\[5ex] = \dfrac{21}{7} + 12 \div \dfrac{8}{5} \\[5ex] = 3 + 12 * \dfrac{5}{8} \\[5ex] = 3 + 3 * \dfrac{5}{2} \\[5ex] = \dfrac{6}{2} + \dfrac{15}{2} \\[5ex] = \dfrac{21}{2} \\[5ex] (b.) (i.) \\[3ex] \underline{\text{Maranda}} \\[3ex] \text{Monthly Earnings} = \$8316 \\[3ex] \text{Expenses} = \dfrac{4}{7} * 8316 = \$4752 \\[5ex] \text{Remainder = Savings} = \$8316 - \$4752 = \$3564 \\[3ex] \text{What % of 8316 is 3564}? \\[3ex] \dfrac{is}{of} = \dfrac{what}{100} \quad\dots\text{Percent-Proportion} \\[5ex] \dfrac{3564}{8316} = \dfrac{what}{100} \\[5ex] what = \dfrac{3564 * 100}{8316} \\[5ex] = 42.85714286 \\[3ex] \approx 42.9\% \quad\dots\text{to 3 significant figures} \\[3ex] (ii.) \\[3ex] \underline{\text{Maranda's Expenses}} \\[3ex] \text{Utility Bills : Food : Personal Items = 5 : 3 : 4} \\[3ex] \text{Sum of ratios} = 5 + 3 + 4 = 12 \\[3ex] \text{Expense on Personal Items} = \dfrac{4}{12} * 4752 \\[5ex] = \$1584 \\[3ex] (iii.) \\[3ex] \text{Monthly Savings} = \$3564 \\[3ex] \text{Annual Savings} = 3564 * 12 = \$42768 \\[3ex] (iv.) \\[3ex] P = \$42768 \\[3ex] r = 5\% \\[5ex] n = 2\text{ years} \\[3ex] A = P\left(1 + \dfrac{r}{100}\right)^n \\[5ex] = 42768 * \left(1 + \dfrac{5}{100}\right)^2 \\[5ex] = 42768 * (1 + 0.05)^2 \\[3ex] = 42768 * 1.05^2 \\[3ex] = 42768 * 1.1025 \\[3ex] = \$47151.72 \\[3ex] \text{Total Interest earned after 2 years} \\[3ex] = A - P \\[3ex] = 47151.72 - 42768 \\[3ex] = \$4383.72 $
