Question paper, Set 10

$100000$ has $5$ zeros, so move the decimal point $5$ places to the left in $6152$.

Write zeros in front to fill the empty places.

$\displaystyle \frac{6152}{100000} = 0.06152$

Answer: $0.06152$

Dividing by $10$ moves the decimal point $1$ place to the left (put zeros in front where needed).

$\displaystyle \frac{0.802}{10} = 0.0802$

Answer: $0.0802$

Multiplying by $10000$ moves the decimal point $4$ places to the right (fill zeros if needed).

$0.0715 \times 10000 = 715.0 = 715$

Answer: $715$

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.

$0060.6000 = 60.6$

Answer: $60.6$

Place values: $3 \to 300$, $6 \to 60$, $7 \to 7$, $\displaystyle 8 \to \frac{8}{10}$, $\displaystyle 4 \to \frac{4}{100}$, $\displaystyle 9 \to \frac{9}{1000}$

$\displaystyle 367.849 = 300 + 60 + 7 + \frac{8}{10} + \frac{4}{100} + \frac{9}{1000}$

Answer: $\displaystyle 300 + 60 + 7 + \frac{8}{10} + \frac{4}{100} + \frac{9}{1000}$

In $2390.84$, the digit $3$ is in the hundreds place.

$\text{Place value of } 3 = 300$

Answer: $300$

1st method (without actual division)

$250 = 2 \times 5 \times 5 \times 5$; pair each $2$ with a $5$ to make $10$s.

$\displaystyle \frac{41}{250} = \frac{41 \times 2 \times 2}{2 \times 5 \times 5 \times 5 \times 2 \times 2}$

$\displaystyle = \frac{164}{10 \times 10 \times 10} = \frac{164}{1000} = 0.164$

Answer: $0.164$

2nd method (division method): divide $29$ by $40$.

$29 < 40$, so write $0.$ and bring down a $0$

Bring down $0$: $290$; $40 \times 7 = 280$, $290 - 280 = 10$

Bring down $0$: $100$; $40 \times 2 = 80$, $100 - 80 = 20$

Bring down $0$: $200$; $40 \times 5 = 200$, $200 - 200 = 0$

Remainder $0$. $\displaystyle \therefore \frac{29}{40} = 0.725$

Answer: $0.725$

$21 = 3 \times 7$ has the factor $3$ (not $2$ or $5$).

So it is a non-terminating recurring decimal.

Answer: Non-terminating recurring decimal

Divide $1$ by $13$.

$1 < 13$, so write $0.$ and bring down a $0$

Bring down $0$: $10 < 13$, so write $0$

Bring down $0$: $100$; $13 \times 7 = 91$, $100 - 91 = 9$

Bring down $0$: $90$; $13 \times 6 = 78$, $90 - 78 = 12$

Bring down $0$: $120$; $13 \times 9 = 117$, $120 - 117 = 3$

Bring down $0$: $30$; $13 \times 2 = 26$, $30 - 26 = 4$

Bring down $0$: $40$; $13 \times 3 = 39$, $40 - 39 = 1$

The remainder $1$ has come again, so the digits $076923$ repeat.

$\displaystyle \therefore \frac{1}{13} = 0.076923076923\ldots = 0.\overline{076923}$

Answer: $0.\overline{076923}$

$\displaystyle 7.5 = \frac{75}{10}$ ($1$ digit after the point, so $10$)

Divide both by $5$: $\displaystyle \frac{75 \div 5}{10 \div 5} = \frac{15}{2}$

$\displaystyle \therefore 7.5 = \frac{15}{2}$

Answer: $\displaystyle \frac{15}{2}$

Let $x = 2.\overline{18} = 2.181818\ldots$

$100x = 218.181818\ldots$

$x = 2.181818\ldots$

$(-)$ $99x = 216$

$\displaystyle x = \frac{216}{99} = \frac{24}{11}$

$\displaystyle \therefore 2.\overline{18} = \frac{24}{11}$

Answer: $\displaystyle \frac{24}{11}$

Let $x = 0.5\overline{36} = 0.5363636\ldots$

$10x = 5.363636\ldots$ (the repeating part starts just after the point)

$1000x = 536.363636\ldots$

$10x = 5.363636\ldots$

$(-)$ $990x = 531$

$\displaystyle x = \frac{531}{990} = \frac{59}{110}$

Check (short method): $\displaystyle \frac{536 - 5}{990} = \frac{531}{990} = \frac{59}{110}$

$\displaystyle \therefore 0.5\overline{36} = \frac{59}{110}$

Answer: $\displaystyle \frac{59}{110}$

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

$\begin{array}{r} 230.500 \\ 17.096 \\ +\;3.840 \\ \hline 251.436 \end{array}$

$\therefore 230.5 + 17.096 + 3.84 = 251.436$

Answer: $251.436$

Group the terms: $(77.7 + 205.39) - (18.064 + 9.85)$

Positive terms: $\begin{array}{r} 77.70 \\ +\;205.39 \\ \hline 283.09 \end{array}$

Negative terms: $\begin{array}{r} 18.064 \\ +\;9.850 \\ \hline 27.914 \end{array}$

Subtract: $\begin{array}{r} 283.090 \\ -\;27.914 \\ \hline 255.176 \end{array}$

$\therefore 77.7 - 18.064 + 205.39 - 9.85 = 255.176$

Answer: $255.176$

Multiply without the points: $1003 \times 46 = 46138$

Decimal places: $3 + 1 = 4$

$1.003 \times 4.6 = 4.6138$

Answer: $4.6138$

$\displaystyle \frac{72.5}{25} = \frac{725}{10 \times 25}$

$725 \div 25 = 29$

$\displaystyle = \frac{29}{10} = 2.9$

Answer: $2.9$

$0.08$ has $2$ digits after the point, so multiply both by $100$:

$\displaystyle \frac{52.8}{0.08} = \frac{52.8 \times 100}{0.08 \times 100} = \frac{5280}{8}$

$5280 \div 8 = 660$

$\therefore 52.8 \div 0.08 = 660$

Answer: $660$

$0.5 \times 0.6 = 0.30 = 0.3$

$0.5 \times 0.02 = 0.010 = 0.01$

$\displaystyle \frac{0.48}{0.6} = \frac{0.48 \times 10}{0.6 \times 10} = \frac{4.8}{6} = 0.8$

Value $= 0.3 - 0.01 + 0.8 = 1.09$

Answer: $1.09$

Let the other decimal be $x$. Then $x \times 3.75 = 7.8$

$\displaystyle x = \frac{7.8}{3.75} = \frac{7.8 \times 100}{3.75 \times 100} = \frac{780}{375}$

$\displaystyle \frac{780}{375} = \frac{78000}{100 \times 375}$; divide $78000$ by $375$:

$375 \times 2 = 750$, $780 - 750 = 30$

Bring down $0$: $300 < 375$, write $0$

Bring down $0$: $375 \times 8 = 3000$, $3000 - 3000 = 0$

$78000 \div 375 = 208$, so $\displaystyle x = \frac{208}{100} = 2.08$

Check: $2.08 \times 3.75 = 7.8$

$\therefore \text{the other decimal} = 2.08$

Answer: $2.08$