NCERT Solutions for Class 9 Maths Exercise 1.3 Question 2
Understanding the Question 🧐
This is a clever question that tests our observation skills. We are given the decimal expansion of &&\frac{1}{7}&& and asked if we can find the decimal expansions for &&\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}$$, and &&\frac{6}{7}&& without actually doing the long and tedious division process. The answer is yes, and the method is quite simple!
You know that &&\frac{1}{7} = 0.\overline{142857}&&. Can you predict what the decimal expansions of &&\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}&& are, without actually doing the long division? If so, how?
The Strategy: Simple Multiplication 📝
The key is to recognize that each of these fractions can be written as a multiple of &&\frac{1}{7}&&.
- &&\frac{2}{7} = 2 \times \frac{1}{7}&&
- &&\frac{3}{7} = 3 \times \frac{1}{7}&&
- …and so on.
So, all we need to do is multiply the given decimal value of &&\frac{1}{7}&& by the respective numerators (2, 3, 4, 5, and 6).
Step-by-Step Predictions
Given: &&\frac{1}{7} = 0.142857142857… = 0.\overline{142857}&&
1. For &&\frac{2}{7}&&:
&& \frac{2}{7} = 2 \times \frac{1}{7} = 2 \times 0.\overline{142857} &&
&& 2 \times 0.142857 = 0.285714 &&
Therefore, &&\frac{2}{7} = 0.\overline{285714}&&
2. For &&\frac{3}{7}&&:
&& \frac{3}{7} = 3 \times \frac{1}{7} = 3 \times 0.\overline{142857} &&
&& 3 \times 0.142857 = 0.428571 &&
Therefore, &&\frac{3}{7} = 0.\overline{428571}&&
3. For &&\frac{4}{7}&&:
&& \frac{4}{7} = 4 \times \frac{1}{7} = 4 \times 0.\overline{142857} &&
&& 4 \times 0.142857 = 0.571428 &&
Therefore, &&\frac{4}{7} = 0.\overline{571428}&&
4. For &&\frac{5}{7}&&:
&& \frac{5}{7} = 5 \times \frac{1}{7} = 5 \times 0.\overline{142857} &&
&& 5 \times 0.142857 = 0.714285 &&
Therefore, &&\frac{5}{7} = 0.\overline{714285}&&
5. For &&\frac{6}{7}&&:
&& \frac{6}{7} = 6 \times \frac{1}{7} = 6 \times 0.\overline{142857} &&
&& 6 \times 0.142857 = 0.857142 &&
Therefore, &&\frac{6}{7} = 0.\overline{857142}&&
Conclusion and Key Points ✅
Yes, it is possible to predict the decimal expansions without actual long division. A key observation from these ncert solutions is that all the answers are made up of the same block of repeating digits (142857). The only difference is the starting point of the repeating cycle. This is a fascinating property of fractions with a denominator of 7.
- The property &&\frac{n}{d} = n \times \frac{1}{d}&& is the key to this shortcut.
- All fractions from &&\frac{1}{7}&& to &&\frac{6}{7}&& have repeating decimals.
- They all share the same six repeating digits: &&1, 4, 2, 8, 5, 7&&.
- The order of the digits remains the same, but the starting point of the cycle changes for each fraction.
FAQ
Q: How can we predict the decimal expansions without performing long division?
A: We can predict them by taking the given decimal value of &&\frac{1}{7}&& (&&0.\overline{142857}&&) and multiplying it by the numerators 2, 3, 4, 5, and 6 respectively.
Q: What is the key observation about the answers for &&\frac{2}{7}, \frac{3}{7}$$, etc.?
A: The key observation is that all the answers use the same block of repeating digits (&&142857&&), but the sequence starts from a different digit for each fraction. This is known as a cyclic pattern.
Q: Why does this multiplication trick work?
A: It works because &&\frac{2}{7}&& is the same as &&2 \times \frac{1}{7}&&. The deeper reason relates to the remainders obtained during division. The sequence of remainders when dividing ‘n’ by 7 is just a shifted version of the remainders when dividing 1 by 7, which causes the digits in the quotient to also be shifted.
Q: Does this method apply to other fractions, like &&\frac{n}{13}&&?
A: Yes, the same principle applies. The fractions &&\frac{n}{13}&& for &&n=1, 2, …, 12&& will also have decimal expansions whose repeating blocks are cyclic shifts of each other.
Q: What is the given decimal expansion of &&\frac{1}{7}&&?
A: The given value is &&\frac{1}{7} = 0.142857142857…&&, which is written as &&0.\overline{142857}&&.
Further Reading
To learn more about the fascinating properties of repeating decimals and rational numbers, you can refer to the official NCERT textbooks.
