Another Peek Beyond the Point | FIO

Question 11

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) $\dfrac{4827}{8}$

Solution

Understand the Question

To convert a fraction to a decimal quickly, convert its denominator into a power of $10$ $10$ ( $10$ $10$ , $100$ $100$ , or $1000$ $1000$ ) by multiplying both the numerator and denominator by an appropriate factor:
- Multiply $5$ by $2$ to make $10$ .
- Multiply $4$ by $25$ to make $100$ .
- Multiply $2$ by $5$ to make $10$ .
- Multiply $8$ by $125$ to make $1000$ .
Then, verify each result using the standard long division method.

(a) Find the quotient of $\dfrac{18}{5}$ by converting the denominator and verify by long division.

Step 1 · Convert Denominator to 10

Multiply numerator and denominator by $2$ : Diagram 1

\begin{aligned} \dfrac{18}{5} &= \dfrac{18 \times 2}{5 \times 2} \\[0.6em] &= \dfrac{36}{10} \\[0.6em] &= 3.6 \end{aligned}

Step 2 · Verify by Long Division

Divide $18$ by $5$ :

\begin{array}{r} 3.6 \\ 5 \overline{\smash{)} 18.0 } \\ -15 \downarrow \\ \hline 30 \\ -30 \\ \hline 0 \end{array}

Answer

(a) $3.6$

(b) Find the quotient of $\dfrac{415}{4}$ by converting the denominator and verify by long division.

Step 1 · Convert Denominator to 100

Multiply numerator and denominator by $25$ : Diagram 2

\begin{aligned} \dfrac{415}{4} &= \dfrac{415 \times 25}{4 \times 25} \\[0.6em] &= \dfrac{10375}{100} \\[0.6em] &= 103.75 \end{aligned}

Step 2 · Verify by Long Division

Divide $415$ by $4$ :

\begin{array}{r} 103.75 \\ 4 \overline{\smash{)} 415.00 } \\ -4 \downarrow \downarrow \\ \hline 01 \\ -0 \downarrow \\ \hline 15 \\ -12 \downarrow \\ \hline 30 \\ -28 \downarrow \\ \hline 20 \\ -20 \\ \hline 0 \end{array}

Answer

(b) $103.75$

(c) Find the quotient of $\dfrac{1217}{2}$ by converting the denominator and verify by long division.

Step 1 · Convert Denominator to 10

Multiply numerator and denominator by $5$ : Diagram 3

\begin{aligned} \dfrac{1217}{2} &= \dfrac{1217 \times 5}{2 \times 5} \\[0.6em] &= \dfrac{6085}{10} \\[0.6em] &= 608.5 \end{aligned}

Step 2 · Verify by Long Division

Divide $1217$ by $2$ :

\begin{array}{r} 608.5 \\ 2 \overline{\smash{)} 1217.0 } \\ -12 \downarrow \downarrow \\ \hline 01 \\ -0 \downarrow \\ \hline 17 \\ -16 \downarrow \\ \hline 10 \\ -10 \\ \hline 0 \end{array}

Answer

(c) $608.5$

(d) Find the quotient of $\dfrac{4827}{8}$ by converting the denominator and verify by long division.

Step 1 · Convert Denominator to 1000

Multiply numerator and denominator by $125$ : Diagram 4

\begin{aligned} \dfrac{4827}{8} &= \dfrac{4827 \times 125}{8 \times 125} \\[0.6em] &= \dfrac{603375}{1000} \\[0.6em] &= 603.375 \end{aligned}

Step 2 · Verify by Long Division

Divide $4827$ by $8$ :

\begin{array}{r} 603.375 \\ 8 \overline{\smash{)} 4827.000 } \\ -48 \downarrow \downarrow \downarrow \\ \hline 02 \\ -0 \downarrow \\ \hline 27 \\ -24 \downarrow \\ \hline 30 \\ -24 \downarrow \\ \hline 60 \\ -56 \downarrow \\ \hline 40 \\ -40 \\ \hline 0 \end{array}

Answer

(d) $603.375$

Common Mistakes

Forgetting to multiply the numerator: Multiplying only the denominator by the factor changes the value of the fraction. Both numerator and denominator must be multiplied by the same number.
Missing zero in the quotient: When bringing down a digit that is smaller than the divisor (e.g., bringing down $1$ in $415 \div 4$ or $2$ in $4827 \div 8$ ), forgetting to place a $0$ in the quotient before bringing down the next digit leads to an incorrect quotient.

More questions in FIO

Recall that a tenth is $0.1$ , a hundredth is $0.01$ , and so on. Find the following products in tenths, hundredths and so on:

(a) $6 \times 4 \text{ tenths} = 24 \text{ tenths}$

(b) $7 \times 0.3$

Find the products:

(a) $27.34 \times 6$

(b) $4.23 \times 3.7$

Thejus needs $1.65 \text{ m}$ of cloth for a shirt. How many metres of cloth are needed for $3$ shirts?

Meenu bought 4 notebooks and 3 erasers. The cost of each book was ₹15.50 and each eraser was ₹2.75. How much did she spend in all?

The thickness of a rupee coin is $1.45 \text{ mm}$ . What is the total height of the cylinder formed by placing 36 rupee coins one over the other? Write the answer in centimeters.

The price of 1 kg of oranges is ₹56.50. What is the price of 2.250 kg of oranges? Can we write 56.50 as 56.5 and 2.250 as 2.25 and multiply? Will we get the same product? Why?

Dwarakanath purchases notebooks at a wholesale price of ₹23.6 per piece and sells each notebook at ₹30/-. How much profit does he make if he sells 50 books in a week?

Given that $18 \times 12 = 216$ , find the products:

(a) $18 \times 1.2$ (b) $18 \times 0.12$ (c) $1.8 \times 1.2$ (d) $0.18 \times 0.12$ (e) $0.018 \times 0.012$ (f) $1.8 \times 12$

In which of the cases above is the product less than 1?

In which of the following multiplications is the product less than 1? Can you find the answer without actually doing the multiplications?

(a) $7 \times 0.6$

(b) $0.7 \times 0.6$

(d) $0.07 \times 0.06$

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 $2.46 \div 0.15$ ?

Q20

A $4\text{ m}$ long wooden block has to be cut into $5$ pieces of equal length. What is the length of each piece?

Q21

If the perimeter of a regular polygon with 12 sides is 208.8 cm, what is the length of its side?

Q22

3 litres of watermelon juice is shared among 8 friends equally. How much watermelon juice will each get? Express the quantity of juice in millilitres.

Q23

A car covers 234.45 km using 12.6 litres of petrol. What is the distance travelled per litre?

Q24

5 kg of flour (aata) was distributed equally among 15 students. How much flour did each student receive?

Q25

A 210 gram packet of peanut chikki costs ₹70.5, while a 110 gram packet of potato chips costs ₹33.25. Which is cheaper?

Q26

Write the decimal number at the arrow mark:

Q27

Shyamala bought 3 kg bananas at ₹30/- per kg. She counted 35 bananas in all. She sells each banana for ₹5/-. How much profit does she make selling all the bananas?

Q28

A teacher placed textbooks that are 2.5 cm thick on a bookshelf. The teacher wanted to place 80 textbooks on the shelf. The bookshelf is 160 cm long. How many books could be placed on the shelf? Was there any space left? If yes, how much?

Q29

Fill in the following blanks appropriately:

(a) $5.5\text{ km} = \underline{\hspace{1.5cm}}\text{ m}$

(b) $35\text{ cm} = \underline{\hspace{1.5cm}}\text{ m}$

(d) $68\text{ g} = \underline{\hspace{1.5cm}}\text{ kg}$

(e) $9.02\text{ m} = \underline{\hspace{1.5cm}}\text{ mm}$

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

Q30

The following problem was set by Sridharacharya in his book, Patiganita. “ $6\dfrac{1}{4}$ is divided by $2\dfrac{1}{2}$ , and $60\dfrac{1}{4}$ is divided by $3\dfrac{1}{2}$ . Tell the quotients separately.” Can you try to solve it by converting the fractions into decimals?