Try to find the decimal expansions of 10/3 and 11/12. What do you observe about the repetition Class 9

Try to find the decimal expansions of 10/3 and 11/12. What do you observe about the repetition Class 9

Question 1.

Try to find the decimal expansions of $\frac{10}{3}$ and $\frac{11}{12}$. What do you observe about the repetition of the digits after the decimal point? Class 9

Answer:

In $\frac{10}{3}$, the digit 3 repeats continuously after the decimal point.

In $\frac{11}{12}$, the digit 6 repeats continuously after some non-repeating digits.

Thus, the decimal expansion is non-non¬terminating and repeats in a pattern.

Question 2.

The decimal expansion of $\frac{p}{q}$ will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why? Class 9

Answer:

The decimal expansion of $\frac{p}{q}$ will be terminating only when the prime factors of q are 2, 5, or both.

This is because we can make the denominator a power of 10.

Since 10 = 2 × 5, any power of 10 is made using only 2's and 5's.

If the denominator has only 2 and/or 5 as prime factors, we can multiply the numerator and denominator by a suitable number to make the denominator: 10,100,1000, ... and hence, the decimal expansion terminates.