Basic Operations Set B

A. $\dfrac{1}{2}$

$$ 0.5=5\hspace{2mm}\text{tenths}=\dfrac{5}{10} $$

$$ 0.5=\dfrac{1}{2} $$

B. $4$

$$ \begin{aligned}{200}\div{50} &= {20\text{ tens}}\div {5\text{ tens}} \\ \\ {{200}\div{50}} &= 4 \end{aligned} $$

B. $30$

Let’s estimate the quotient by rounding the factors to a friendly number.

A friendly number is a number that might be easy for us to add, subtract, multiply and divide.

Common friendly numbers are multiples of $2$, $5$ and $10$.

$178.4$ is close to $180$.

$6$ is already a friendly number.

$$ 180 \div 6=30 $$

$$ 178.4 \div 6 \approx 30 $$

B. $4$

First, Don multiplies $5 \times 8 = 40$.

Don places the $0$ ones in the ones column and carries the $4$ tens to the tens column.

$$ \begin{aligned} \overset{{}}4\overset{4}1\overset{}{8}&\\ \underline{{} \times \phantom{~~~~} 5}&\\ 0& \quad {5} \times {8} = {40} \end{aligned} $$

Don should replace $y$ with $4$.

B. $8\dfrac{3}{4}$

First, let’s rewrite $1\dfrac{3}{4}$ as a fraction.

$$ 1{\dfrac{3}{4}} = {1} +{\dfrac{3}{4}} $$

$$ \phantom{1\dfrac{3}{4} }= {\dfrac{4}{4}} + {\dfrac{3}{4}} $$

$$ \phantom{1\dfrac{3}{4} }= \dfrac{{4} +3}{4} $$

$$ \phantom{1\dfrac{3}{4} }= \dfrac{7}{4} $$

Then, we can multiply.

$$ \phantom{=} 5 \times \dfrac{7}{4} $$

$$ =\dfrac{5\times 7}{4} $$

$$ =\dfrac{35}{4} $$

The product, in lowest terms, is $\dfrac{35}{4}$.

We can also write this as $8\dfrac{3}{4}$.

C. $\dfrac{33}{4}$

$$\dfrac{4}{3} = \dfrac{11}{k}$$

Multiply both sides by $k$.

$$ {k} \times \dfrac{4}{3} = \dfrac{11}{k} \times {k} $$

$$ \dfrac{4}{3}{k} = 11 $$

Multiply both sides by ${\dfrac{3}{4}}$.

$$ \dfrac{4}{3} {k} \times {\dfrac{3}{4}} = 11 \times {\dfrac{3}{4}} $$

$$ {k} = \dfrac{11 \times {3}}{{4}} $$

$$ k = \dfrac{33}{4} $$

A. $1.25$

There are many ways to solve this problem. Let’s see three ways we could multiply.

1. Place value strategy

We can think in terms of hundredths:

$$ \begin{aligned} &\phantom{{}={}}5\times 0.25\\ &=5\times25\text{ hundredths}\\ &= 125\text{ hundredths}\\ &=1.25 \end{aligned} $$

2. Estimation strategy

We can multiply the digits, then use estimation to place the decimal point.

$$ 5\times25=125 $$

The value $0.25$ is equal to $\dfrac{1}{4}$.

$\dfrac{1}{4}$ of $5$ is between $1$ and $2$, so $5\times 0.25$ is between $1$ and $2$.

$$ 5\times 0.25=1.25 $$

3. Fraction multiplication strategy

Decimals are a kind of fraction, so we can use fraction multiplication.

$$ \begin{aligned} &\phantom{{}={}}{5}\times{0.25}\\ &={\dfrac{5}{1}} \times {\dfrac{25}{100}}\\ &= \dfrac{{5} \times {25}}{{1} \times {100}}\\ &=\dfrac{125}{100}\\ &=1.25 \end{aligned} $$

The answer

$$ 1.25=5 \times 0.25 $$

A. Yes

$156,970$ is divisible by $5$ if it can be divided by $5$ without leaving a remainder.

A number is divisible by $5$ if the last digit is a $0$ or a $5$.

The last digit of $156,970$ is $0$, so yes $156,970$ is divisible by $5$.

C. $361$ units

Formula for the circumference of a circle

$$ \text{Circumference}= 2\pi r = \pi d $$

Solving for the diameter

Let’s substitute in the circumference, then solve for the diameter.

$$ \begin{aligned} 1{,}133.54 &= \pi d \\ 1{,}133.54&\approx 3.14 d\\ 361&\approx d \end{aligned} $$

The answer

The diameter is approximately $361$ units.

B. $83.5$

We can group the whole numbers together and the decimals together.

$$ \begin{aligned} &({55} + {28}) + ({0.3} + {0.2})\\ &={83} + {0.5}\\ &=83.5 \end{aligned} $$

$$ 83.5 = 55.3 + 28.2 $$