Evaluate the following using the information : (a) (b) (c) (d)

Understand the Question We are given the basic arithmetic relationship: From this multiplication fact, the corresponding division facts are: To evaluate decimals with these base digits, express decimals as fractions or division by powers of (), apply the known fact, and adjust the decimal point accordingly. Part (a) Step 1 · Calculate Given . Answer > (a) Part (b) Step 1 · Calculate Given . Answer > (b) Part (c) Step 1 · Calculate Given . Answer > (c) Part (d) Step 1 · Calculate Given . Answer > (d) Common Mistakes Missing Leading Zeros in Multiplication: In part (d), has decimal places. Writing instead of inserting a placeholder zero to get is a frequent mistake. Miscounting Decimal Places in Division: In division, the number of decimal places in the quotient is . For example, in (b), decimal places, giving (an integer).

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Question 17

Evaluate the following using the information $156 \times 12 = 1872$ :

(a) $15.6 \times 1.2 = \underline{\quad\quad\quad\quad}$

(b) $187.2 \div 1.2 = \underline{\quad\quad\quad\quad}$

(d) $0.156 \times 0.12 = \underline{\quad\quad\quad\quad}$

Solution

Understand the Question

We are given the basic arithmetic relationship: $156 \times 12 = 1872$
From this multiplication fact, the corresponding division facts are: $1872 \div 12 = 156 \quad \text{and} \quad 1872 \div 156 = 12$
To evaluate decimals with these base digits, express decimals as fractions or division by powers of $10$ ( $10, 100, 1000, \dots$ ), apply the known fact, and adjust the decimal point accordingly.

(a) $15.6 \times 1.2 = \underline{\quad\quad\quad\quad}$

Step 1 · Calculate $15.6 \times 1.2$

Given $156 \times 12 = 1872$ .

Diagram 1

\begin{aligned} 15.6 \times 1.2 &= (156 \div 10) \times (12 \div 10) \\[0.6em] &= (156 \times 12) \div (10 \times 10) \\[0.6em] &= 1872 \div 100 \\[0.6em] &= 18.72 \end{aligned}

Answer

(a) $18.72$

(b) $187.2 \div 1.2 = \underline{\quad\quad\quad\quad}$

Step 1 · Calculate $187.2 \div 1.2$

Given $156 \times 12 = 1872 \implies 1872 \div 12 = 156$ .

Diagram 2

\begin{aligned} 187.2 \div 1.2 &= (1872 \div 10) \div (12 \div 10) \\[0.6em] &= (1872 \div 10) \times (10 \div 12) \\[0.6em] &= 1872 \div 12 \\[0.6em] &= 156 \end{aligned}

Answer

(b) $156$

(c) $18.72 \div 15.6 = \underline{\quad\quad\quad\quad}$

Step 1 · Calculate $18.72 \div 15.6$

Given $156 \times 12 = 1872 \implies 1872 \div 156 = 12$ .

\begin{aligned} 18.72 \div 15.6 &= (1872 \div 100) \div (156 \div 10) \\[0.6em] &= (1872 \div 100) \times (10 \div 156) \\[0.6em] &= (1872 \div 156) \times (10 \div 100) \\[0.6em] &= 12 \times (1 \div 10) \\[0.6em] &= 12 \times 0.1 \\[0.6em] &= 1.2 \end{aligned}

Answer

(c) $1.2$

(d) $0.156 \times 0.12 = \underline{\quad\quad\quad\quad}$

Step 1 · Calculate $0.156 \times 0.12$

Given $156 \times 12 = 1872$ .

Diagram 4

\begin{aligned} 0.156 \times 0.12 &= (156 \div 1000) \times (12 \div 100) \\[0.6em] &= (156 \times 12) \div (1000 \times 100) \\[0.6em] &= 1872 \div 100000 \\[0.6em] &= 0.01872 \end{aligned}

Answer

(d) $0.01872$

Common Mistakes

Missing Leading Zeros in Multiplication: In part (d), $0.156 \times 0.12$ has $3 + 2 = 5$ decimal places. Writing $0.1872$ instead of inserting a placeholder zero to get $0.01872$ is a frequent mistake.
Miscounting Decimal Places in Division: In division, the number of decimal places in the quotient is $(\text{places in dividend}) - (\text{places in divisor})$ . For example, in (b), $1 - 1 = 0$ decimal places, giving $156$ (an integer).

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Q10

Multiplying the following numbers by $10$ , $100$ and $1000$ to complete the table.

Q11

Find the quotient by converting the denominator into 1, 10, 100 or 1000 and verify the solution by the long division method (division by place value).

(a) $\dfrac{18}{5}$ (b) $\dfrac{415}{4}$ (c) $\dfrac{1217}{2}$ (d) $\dfrac{4827}{8}$

Q12

Choose the correct answer:

(a) $\dfrac{1526}{4} =$

(i) 38.15

(ii) 380.15

(iii) 381.5

(iv) 381.05

(b) $\dfrac{3567}{8} =$

(i) 4458.75

(ii) 44.5875

(iii) 445.875

(iv) 4458.75

Q13

What is the quotient?

(a) $132 \div 4 =$

(b) $13.2 \div 4 =$

(d) $0.132 \div 4 =$

Q14

What is the quotient?

(a) $126 \div 8 =$

(b) $12.6 \div 8 =$

(d) $0.126 \div 8 =$

(e) $0.0126 \div 8 =$

Q15

Express the following fractions in decimal form:

(a) $\dfrac{2}{5}$

(b) $\dfrac{13}{4}$

(d) $\dfrac{5}{8}$

Q16

Find the quotients:

(a) $24.86 \div 1.2$

(b) $5.728 \div 1.52$

Q17

Evaluate the following using the information $156 \times 12 = 1872$ :

(a) $15.6 \times 1.2 = \underline{\quad\quad\quad\quad}$

(b) $187.2 \div 1.2 = \underline{\quad\quad\quad\quad}$

(d) $0.156 \times 0.12 = \underline{\quad\quad\quad\quad}$

Q18

Evaluate the following:

(a) $25 \div \underline{\quad\quad} = 0.025$

(b) $25 \div \underline{\quad\quad} = 250$

(d) $25 \div 10 = 25 \times \underline{\quad\quad}$

(e) $25 \div 0.10 = 25 \times \underline{\quad\quad}$

(f) $25 \div 0.01 = 25 \times \underline{\quad\quad}$

Q19

Find the quotient:

(a) $2.46 \div 1.5 =$

(b) $2.46 \div 0.15 =$

Is the quotient obtained in $24.6 \div 1.5$ the same as the quotient obtained in $2.46 \div 0.15$ ?

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Write the decimal number at the arrow mark:

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Q29

Fill in the following blanks appropriately:

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(f) $125.5\text{ ml} = \underline{\hspace{1.5cm}}\text{ l}$

Q30

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