Another Peek Beyond the Point | IT

Question 16

Will the quotient be always greater than the dividend when the divisor is a decimal? Try it out with different values of the divisor.

Describe the relationship between the dividend, divisor, and the quotient. Create a table for capturing this relationship in different situations, like we did for multiplication.

Solution

Understand the Question

The question explores what happens to the quotient when dividing a number by a decimal divisor.
A common misconception is that dividing by a decimal always results in a larger quotient. However, decimal numbers can be:
- Less than 1 (e.g., $0.5, 0.2$ ), where dividing makes the number larger ( $\text{Quotient} > \text{Dividend}$ ).
- Greater than 1 (e.g., $2.5, 1.25$ ), where dividing makes the number smaller ( $\text{Quotient} < \text{Dividend}$ ).
By testing different values, we can establish the exact relationship between the dividend, divisor, and quotient.

Step 1 · Test with a Decimal Divisor Less Than 1

Let Dividend $= 10$ and Divisor $= 0.5$ :

\begin{aligned} 10 \div 0.5 &= 100 \div 5 \\[0.6em] &= 20 \end{aligned}

Here, the quotient ( $20$ ) is greater than the dividend ( $10$ ).

Step 2 · Test with a Decimal Divisor Greater Than 1

Let Dividend $= 10$ and Divisor $= 2.5$ :

\begin{aligned} 10 \div 2.5 &= 100 \div 25 \\[0.6em] &= 4 \end{aligned}

Here, the quotient ( $4$ ) is smaller than the dividend ( $10$ ).

Step 3 · Describe the General Relationship

From the tests, the quotient is not always greater than the dividend when dividing by a decimal. The relationship depends on the value of the divisor relative to $1$ :

When $\text{Divisor} < 1$ , $\text{Quotient} > \text{Dividend}$ .
When $\text{Divisor} = 1$ , $\text{Quotient} = \text{Dividend}$ .
When $\text{Divisor} > 1$ , $\text{Quotient} < \text{Dividend}$ .

Step 4 · Tabulate the Relationship in Different Situations

\begin{array}{|c|c|c|c|} \hline \text{Dividend} & \text{Divisor} & \text{Quotient} & \text{Relationship (Quotient vs. Dividend)} \\ \hline 10 & 0.5 & 20 & \text{Quotient is greater than Dividend} \\ \hline 10 & 1 & 10 & \text{Quotient is equal to Dividend} \\ \hline 10 & 2.5 & 4 & \text{Quotient is smaller than Dividend} \\ \hline 5 & 0.2 & 25 & \text{Quotient is greater than Dividend} \\ \hline 5 & 1 & 5 & \text{Quotient is equal to Dividend} \\ \hline 5 & 1.25 & 4 & \text{Quotient is smaller than Dividend} \\ \hline \end{array}

Answer

No, the quotient is not always greater than the dividend when the divisor is a decimal.

If $\text{Divisor} < 1$ , then $\text{Quotient} > \text{Dividend}$
If $\text{Divisor} = 1$ , then $\text{Quotient} = \text{Dividend}$
If $\text{Divisor} > 1$ , then $\text{Quotient} < \text{Dividend}$

Common Mistakes

Assuming all decimals are less than 1: Forgetting that numbers like $2.5$ or $1.25$ are also decimals; because they are $> 1$ , dividing by them makes the quotient smaller than the dividend.
Generalizing whole-number division rules: Assuming that division always produces a smaller number, which only holds true when the divisor is strictly greater than $1$ .

More questions in IT

Jonali and Pallabi play a game. Jonali says a fraction and Pallabi gives the equivalent decimal. Write Pallabi's answer in the blank spaces.

Jonali goes to the market to buy spices. She purchases $50\text{ g}$ of Cinnamon, $100\text{ g}$ of Cumin seeds, $25\text{ g}$ of Cardamom and $250\text{ g}$ of Pepper. Express each of the quantities in kilograms by writing them in terms of fractions as well as decimals.

Write the following fractions as a sum of fractions and also as decimals:

Can you give a simple rule to divide any number by a number of the form 1 followed by zeroes — 10, 100, 1000, etc.? For example, $\dfrac{123}{10}$ , $\dfrac{24}{100}$ or $\dfrac{678}{1000}$ ? Look for a pattern in the previous problems.

Can the product of two decimals be a natural number?

Can the product of a decimal and a natural number be a natural number?

Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Suppose we know that $596 \times 248 = 147808$ , can you immediately write down the product of $5.96 \times 24.8$ ?

Context: Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Q. By looking at the above examples, can you frame a rule to multiply two decimals?

Q10

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Q11

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Q12

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Q13

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Q14

Context: Let us consider the number $142857$ that arose when dividing $1$ by $7$ . Multiply $142857$ by numbers from $1$ to $6$ .

Q. What are the products? What do you notice?

Multiply $142857$ by $7$ . What do you observe?

Q15

To find one such number, you can find $1 \div 17$ in decimal, and use the repeating block of digits.

Q16

Will the quotient be always greater than the dividend when the divisor is a decimal? Try it out with different values of the divisor.

Describe the relationship between the dividend, divisor, and the quotient. Create a table for capturing this relationship in different situations, like we did for multiplication.

Q17

What pattern do you observe? Why are 2 and 5 related in this way?

Q18

Do you know which month has this extra day?

Q19

With this new scheme of adding one extra day every 4th year, what is the number of days in 100 calendar years? Can you write an expression to calculate that number?

Q20

How many years are divisible by 4 in 100 years?

Q21

Can you form different expressions for the same question?

Q22

Context: The calendar designers decided that they will not add 1 extra day in every hundredth year.

Q. Can you write an expression for the number of days in 100 calendar years with this new adjustment?

Q23

With this final scheme of leap years can you calculate the number of calendar days in 10,000 years and the number of actual days the Earth will take to make 10,000 revolutions around the Sun? What is the difference? If there is a big difference, can you suggest a way to fix this problem?

Q24

Do you wonder how people figured out that the Earth completes one revolution around the Sun in exactly 364.2422 days?