From Zero to Engineer

Arithmetic - review exercises II

Ratios of the ingredients of a mixture, converting fractions to percentages and back, calculations correct to four significant figures, repeating decimals and operations on powers.

The second part of the review covers what came after fractions in this course: ratios, percentages, accuracy of a calculation, and powers. The exercises are short, but each one tests a different skill - from finding the proportions of a mixture to writing a repeating decimal as a common fraction. In calculations with several brackets it pays to work from the inside out and write down every step, because that is where mistakes happen.

Arithmetic - review exercises II

In each of the following cases, the proportions of a mixture are given. Find the ratio for each one.

a) 3/4 of ingredient A, 1/4 of ingredient B.

3 : 1

A : B

b) 2/3 of ingredient P, 1/15 of ingredient Q, and remainder of ingredient R.

\frac{2}{3} = \frac{2}{3} \cdot \frac{5}{5} = \frac{10}{15}

\frac{15}{15} - \left(\frac{1}{15} + \frac{10}{15}\right) = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}

10 : 1 : 4

P : Q : R

c) 1/5 of ingredient R, 3/5 of ingredient S, 1/6 of ingredient T, and remainder of ingredient U.

\frac{1}{5} = \frac{1}{5} \cdot \frac{6}{6} = \frac{6}{30}

\frac{3}{5} = \frac{3}{5} \cdot \frac{6}{6} = \frac{18}{30}

\frac{1}{6} = \frac{1}{6} \cdot \frac{5}{5} = \frac{5}{30}

\frac{30}{30} - \left(\frac{6}{30} + \frac{18}{30} + \frac{5}{30}\right) = \frac{30}{30} - \frac{29}{30} = \frac{1}{30}

6 : 18 : 5 : 1

R : S : T : U

Express 3/5 as a percentage.

\frac{3}{5} \cdot \frac{20}{20} = \frac{60}{100} = 60\%

Express 16% as a fraction in its simplest form.

16\% = \frac{16}{100} = \frac{16 : 4}{100 : 4} = \frac{4}{25}

Find 17.5% of 12.50.

17.5\% \cdot 12.50 = \frac{17.5}{100} \cdot 12.50 = \frac{17.5}{8} = 2.1875

100 : 12.5 = 8

Evaluate each of the following expressions correct to four significant figures and three decimal places.

\text{a)}\quad 13.6 \cdot 25.8 : 4.2 \approx 83.54 \approx 83.543

\text{b)}\quad 13.6 : 4.2 \cdot 25.8 \approx 83.54 \approx 83.543

\begin{aligned}\text{c)}\quad 9.1(17.43 + 7.2(8.6 - 4.1^{2} \cdot 3.1)) &= 9.1(17.43 + 7.2(8.6 - 16.81 \cdot 3.1)) \\&= 9.1(17.43 + 7.2(8.6 - 52.111)) \\&= 9.1(17.43 + 7.2 \cdot (-43.511)) \\&= 9.1(17.43 - 313.2792) \\&= 9.1 \cdot (-295.8492) = -2692.22772 \\&\approx -2692 \approx -2692.228\end{aligned}

\begin{aligned}\text{d)}\quad -8.4((6.3 \cdot 9.1 + 2.2^{1.3}) - (4.1^{-3.1} : 3.3^{3} - 5.4)) &= \\&\hspace{-6em} = -8.4((6.3 \cdot 9.1 + 2.787) - (0.0126 : 35.937 - 5.4)) \\&\hspace{-6em} = -8.4((57.33 + 2.787) - (0.00035 - 5.4)) \\&\hspace{-6em} = -8.4(60.117 + 5.39965) = -8.4 \cdot 65.51665 = -550.33986 \\&\hspace{-6em} \approx -550.3 \approx -550.340\end{aligned}

Convert each of the following common fractions to decimal form, correct to three decimal places.

\tfrac{3}{17} = 0.176

\tfrac{-2}{15} = -0.133

\tfrac{17}{3} = 5.667

\tfrac{-24}{11} = -2.182

Write each of the following numbers in its simplest form.

\text{a)}\quad 6.7777\ldots = 6{,}(7)

\text{b)}\quad 0.01001001001\ldots = 0{,}(010)

Convert each of the following decimal numbers to a common fraction in its simplest form.

\text{a)}\quad 0.4 = \frac{4}{10} = \frac{2}{5}

\text{b)}\quad 3.68 = \frac{368}{100} = \frac{92}{25}

c) 1.(4)

1{,}(4) = \tfrac{13}{9}

1{,}(4) \cdot 10 = 14{,}(4)

1{,}(4) \cdot 10 - 1{,}(4) = 14{,}(4) - 1{,}(4)

1{,}(4) \cdot 9 = 13 \quad / : 9

\text{d)}\quad -6.1 = -\frac{61}{10}

Write each of the following numbers as a single number raised to a power.

\text{a)}\quad 2^{9} \cdot 2^{2} = 2^{9+2} = 2^{11}

\text{b)}\quad 6^{2} : 5^{2} = \left(\frac{6}{5}\right)^{2}

\text{c)}\quad \left((-4)^{4}\right)^{-4} = (-4)^{4 \cdot (-4)} = (-4)^{-16}

\text{d)}\quad \left(3^{-5}\right)^{0} = 3^{-5 \cdot 0} = 3^{0} = 1

Frequently asked questions

How do you find the ratio of the ingredients of a mixture?

Bring all the fractions to a common denominator and work out the remainder as the whole minus the sum of the others. The numerators then give the ratio directly: 2/3, 1/15 and the remainder are 10/15, 1/15 and 4/15, that is 10 : 1 : 4.

How do you convert a fraction to a percentage?

Scale it so that the denominator becomes 100 - the numerator is then the percentage. For 3/5 multiply the numerator and denominator by 20 to get 60/100, that is 60%.

What is the difference between four significant figures and three decimal places?

Significant figures are counted from the first non-zero digit, decimal places from the point. That is why −2692.22772 to four significant figures is −2692, while to three decimal places it is −2692.228.

How do you turn the repeating decimal 1.(4) into a common fraction?

Multiply it by 10 to shift one period and subtract the original number: 1.(4) · 10 − 1.(4) gives 13, that is 1.(4) · 9 = 13. Hence 1.(4) = 13/9.

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Arithmetic - review exercises

Arithmetic - review exercises III