Let a and b be two non-zero rational numbers such that a + 1/b = 0. Without assigning any numerical values Class 9

Let a and b be two non-zero rational numbers such that a + 1/b = 0. Without assigning any numerical values Class 9

Question 1.

Let a and b be two non-zero rational numbers such that a + $\frac{1}{b}$ = 0. Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer. Class 9

Solution:

Given,

a + $\frac{1}{b}$ = 0

∴ a = $-\frac{1}{b}$

Now, multiplying both sides by b, we get

ab = -1

Since ab = -1 and -1 is a negative rational number.

∴ ab is negative.

Question 2.

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4^th decimal place. Show that such a number can be written in the form $\frac{p}{10^4}$, where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by 2⁴ or 5⁴? Give reasons. Class 9

Solution:

Let the rational number be x.

A rational number whose decimal expansion ends at the 4th decimal place is of the form

x = 0.abcd.

which can be written as, x = $\frac{a b c d}{10000}=\frac{a b c d}{10^4}$

Thus, such a number can be written in the form x = $\frac{p}{10^4}$, where p is an integer not divisible by 10.

If p were divisible by 10, then a factor of 10 would be cancelled from the numerator and denominator, and the decimal expansion would end before the 4^th decimal place.

Also,

10⁴ = 2⁴ × 5⁴

When the fraction is written in lowest form, common factors of p with 2 or 5 may get cancelled.

Therefore, it is not necessary that the denominator in lowest form must be divisible by 2⁴ or 5⁴.

For example,

$\frac{3756}{10000}=\frac{939}{2500}$

Here, the denominator is 2500, which is not divisible by 2⁴.

Hence, the denominator in lowest form need not be divisible by both 2⁴ and 5⁴.