Question

The rational number that lies between \[\dfrac{{ - 3}}{4}\] and \[\dfrac{{ - 7}}{4}\] on the number line is
A.3
B.2
C.1
D.\[ - 1\]

Answer 1

Hint: To solve this question, we will first convert the given rational numbers into the decimal form. We will then analyze all the options by drawing the number line to see which number lies between the given numbers.

Complete step-by-step answer:
We will first simplify the rational number \[\dfrac{{ - 3}}{4}\].
To simplify it we will divide the numerator with the denominator. On doing so we get,
\[\dfrac{{ - 3}}{4} = - 0.75\]
We will now simplify the rational number \[\dfrac{{ - 7}}{4}\].
Dividing \[ - 7\] by 4, we get
\[\dfrac{{ - 7}}{4} = - 1.75\]
Let us see whether 3 lies between \[\dfrac{{ - 3}}{4}\] and
\[\dfrac{{ - 7}}{4}\] on the number line. So, we will locate 3, \[ - 0.75\], and \[ - 1.75\] on the number line. We will then see if 3 lies between the given numbers on the number line or not.
Locate \[ - 0.75\] and \[ - 1.75\] on the number line. On doing so we get,

We will now locate 3 on the number line.

Now we can see that the blue dot does not lie between the two red dots. This means that 3 does not lie between \[ - 0.75\] and \[ - 1.75\] on the number line.
Now let us see whether 2 lies between \[\dfrac{{ - 3}}{4}\] and \[\dfrac{{ - 7}}{4}\] on the number line. To do so we will locate 2, \[ - 0.75\], and \[ - 1.75\] on the number line. We will then see if 2 lies between the given numbers on the number line or not.
Locating 2, \[ - 0.75\] and \[ - 1.75\] on the number line, we get

Now we can see that the blue dot does not lie between the two red dots. This means that 2 does not lie between \[ - 0.75\] and \[ - 1.75\] on the number line.
Let us see whether 1 lies between \[\dfrac{{ - 3}}{4}\]and \[\dfrac{{ - 7}}{4}\] on the number line. Now, we will locate 1, \[ - 0.75\] and \[ - 1.75\] on the number line. We will then see if 1 lies between the given numbers on the number line or not.
Locating 1, \[ - 0.75\] and \[ - 1.75\] on the number line, we get

Now we can see that the blue dot does not lie between the two red dots. This means that 1 does not lie between \[ - 0.75\] and \[ - 1.75\] on the number line.
Let us see whether \[ - 1\] lies between \[\dfrac{{ - 3}}{4}\] and \[\dfrac{{ - 7}}{4}\] on the number line. We will locate \[ - 1\], \[ - 0.75\] and \[ - 1.75\] on the number line. We will then see if \[ - 1\] lies between the given numbers on the number line or not.
Locate \[ - 1\], \[ - 0.75\] and \[ - 1.75\] on the number line, we get

Now we can see that the blue dot lies between the two red dots. This means that \[ - 1\] lies between \[ - 0.75\] and \[ - 1.75\] on the number line.
Hence, the correct answer is option (D).

Note: To solve this question we need to know the concept of the rational numbers.
Rational numbers are the numbers which are expressed in the form of a fraction \[\dfrac{p}{q}\], where \[q \ne 0\]. The difference between fractions and rational numbers, however, is that rational numbers are both positive and negative, unlike the fractions. Also, when the denominator is 1, rational numbers become integers. Thus, it is said that all integers are rational numbers, but all rational numbers are not integers.