Question paper, Set 6

$1000$ has $3$ zeros, so move the decimal point $3$ places to the left in $45213$.

$\displaystyle \frac{45213}{1000} = 45.213$

Answer: $45.213$

Dividing by $1000$ moves the decimal point $3$ places to the left (put zeros in front where needed).

$\displaystyle \frac{0.047}{1000} = 0.000047$

Answer: $0.000047$

Multiplying by $1000$ moves the decimal point $3$ places to the right (fill zeros if needed).

$0.57 \times 1000 = 570.0 = 570$

Answer: $570$

Zeros in front of the whole number part and zeros at the end of the decimal part do not change the value.

Zeros in the middle must stay.

$000.00700 = 0.007$

Answer: $0.007$

Thousands $7$, hundreds $3$, tens $2$, ones $9$, tenths $5$, hundredths $1$, thousandths $6$

Put $0$ in any place that is missing.

$\displaystyle 7000 + 300 + 20 + 9 + \frac{5}{10} + \frac{1}{100} + \frac{6}{1000} = 7329.516$

Answer: $7329.516$

In $9127.36$, the digit $1$ is in the hundreds place.

$\text{Place value of } 1 = 100$

Answer: $100$

1st method (without actual division)

$50 = 2 \times 5 \times 5$; pair each $2$ with a $5$ to make $10$s.

$\displaystyle \frac{21}{50} = \frac{21 \times 2}{2 \times 5 \times 5 \times 2}$

$\displaystyle = \frac{42}{10 \times 10} = \frac{42}{100} = 0.42$

Answer: $0.42$

2nd method (division method): divide $37$ by $8$.

$8 \times 4 = 32$, $37 - 32 = 5$

Bring down $0$: $50$; $8 \times 6 = 48$, $50 - 48 = 2$

Bring down $0$: $20$; $8 \times 2 = 16$, $20 - 16 = 4$

Bring down $0$: $40$; $8 \times 5 = 40$, $40 - 40 = 0$

Remainder $0$. $\displaystyle \therefore \frac{37}{8} = 4.625$

Answer: $4.625$

$75 = 3 \times 5 \times 5$ has the factor $3$ (not $2$ or $5$).

So it is a non-terminating recurring decimal.

Answer: Non-terminating recurring decimal

Divide $7$ by $13$.

$7 < 13$, so write $0.$ and bring down a $0$

Bring down $0$: $70$; $13 \times 5 = 65$, $70 - 65 = 5$

Bring down $0$: $50$; $13 \times 3 = 39$, $50 - 39 = 11$

Bring down $0$: $110$; $13 \times 8 = 104$, $110 - 104 = 6$

Bring down $0$: $60$; $13 \times 4 = 52$, $60 - 52 = 8$

Bring down $0$: $80$; $13 \times 6 = 78$, $80 - 78 = 2$

Bring down $0$: $20$; $13 \times 1 = 13$, $20 - 13 = 7$

The remainder $7$ has come again, so the digits $538461$ repeat.

$\displaystyle \therefore \frac{7}{13} = 0.538461538461\ldots = 0.\overline{538461}$

Answer: $0.\overline{538461}$

$\displaystyle 3.06 = \frac{306}{100}$ ($2$ digits after the point, so $100$)

Divide both by $2$: $\displaystyle \frac{306 \div 2}{100 \div 2} = \frac{153}{50}$

$\displaystyle \therefore 3.06 = \frac{153}{50}$

Answer: $\displaystyle \frac{153}{50}$

Let $x = 3.\overline{15} = 3.151515\ldots$

$100x = 315.151515\ldots$

$x = 3.151515\ldots$

$(-)$ $99x = 312$

$\displaystyle x = \frac{312}{99} = \frac{104}{33}$

$\displaystyle \therefore 3.\overline{15} = \frac{104}{33}$

Answer: $\displaystyle \frac{104}{33}$

Let $x = 0.08\overline{3} = 0.08333333\ldots$

$100x = 8.333333\ldots$ (the repeating part starts just after the point)

$1000x = 83.333333\ldots$

$100x = 8.333333\ldots$

$(-)$ $900x = 75$

$\displaystyle x = \frac{75}{900} = \frac{1}{12}$

Check (short method): $\displaystyle \frac{83 - 8}{900} = \frac{75}{900} = \frac{1}{12}$

$\displaystyle \therefore 0.08\overline{3} = \frac{1}{12}$

Answer: $\displaystyle \frac{1}{12}$

Write the numbers one below the other with the decimal points in a line (fill zeros):

$\begin{array}{r} 2.394 \\ 15.870 \\ +\;40.500 \\ \hline 58.764 \end{array}$

$\therefore 2.394 + 15.87 + 40.5 = 58.764$

Answer: $58.764$

Group the terms: $(68.53 + 432.1) - (7.906 + 51.48)$

Positive terms: $\begin{array}{r} 68.53 \\ +\;432.10 \\ \hline 500.63 \end{array}$

Negative terms: $\begin{array}{r} 7.906 \\ +\;51.480 \\ \hline 59.386 \end{array}$

Subtract: $\begin{array}{r} 500.630 \\ -\;59.386 \\ \hline 441.244 \end{array}$

$\therefore 68.53 - 7.906 + 432.1 - 51.48 = 441.244$

Answer: $441.244$

Multiply without the points: $413 \times 22 = 9086$

Decimal places: $2 + 1 = 3$

$4.13 \times 2.2 = 9.086$

Answer: $9.086$

$\displaystyle \frac{0.735}{7} = \frac{735}{1000 \times 7}$

$735 \div 7 = 105$

$\displaystyle = \frac{105}{1000} = 0.105$

Answer: $0.105$

$1.2$ has $1$ digit after the point, so multiply both by $10$:

$\displaystyle \frac{43.2}{1.2} = \frac{43.2 \times 10}{1.2 \times 10} = \frac{432}{12}$

$432 \div 12 = 36$

$\therefore 43.2 \div 1.2 = 36$

Answer: $36$

$\displaystyle \frac{3}{4} \times 0.8 = \frac{2.4}{4} = 0.6$

$\displaystyle \frac{1}{4} \times 0.8 = \frac{0.8}{4} = 0.2$

$\displaystyle \frac{0.12}{0.4} = \frac{0.12 \times 10}{0.4 \times 10} = \frac{1.2}{4} = 0.3$

Value $= 0.6 + 0.2 - 0.3 = 0.5$

Answer: $0.5$

Let the other decimal be $x$. Then $x \times 6.08 = 13.072$

$\displaystyle x = \frac{13.072}{6.08} = \frac{13.072 \times 100}{6.08 \times 100} = \frac{1307.2}{608}$

$\displaystyle \frac{1307.2}{608} = \frac{130720}{100 \times 608}$; divide $130720$ by $608$:

$608 \times 2 = 1216$, $1307 - 1216 = 91$

Bring down $2$: $608 \times 1 = 608$, $912 - 608 = 304$

Bring down $0$: $608 \times 5 = 3040$, $3040 - 3040 = 0$

$130720 \div 608 = 215$, so $\displaystyle x = \frac{215}{100} = 2.15$

Check: $2.15 \times 6.08 = 13.072$

$\therefore \text{the other decimal} = 2.15$

Answer: $2.15$