Reduce to the standard form: 48 − 84

Given, 48 − 84 Method 1 : 48 − 84 = 48 − 84 ÷ - 2 - 2 = - 24 42 ÷ 2 2 = - 12 21 ÷ 3 3 = - 4 7 [ Dividing by - 2 , 2 and 3 successively ] Method 2 : Steps to find standard form: Find the HCF of numerator and denominator of the fraction. Divide the numerator and denominator by the produced HCF. Write the simplified fraction of the given fraction. The HCF of 48 , 84 is 12 . Thus, we can get its standard form by dividing it by - 12 . So, 48 - 84 ÷ - 12 - 12 = - 4 7 Hence, the required answer is as shown above.

Reduce to the standard form: - 18 - 42

Given, - 18 - 42 Method 1 : - 18 - 42 = - 18 - 42 ÷ - 2 - 2 = 9 21 ÷ 3 3 = 3 7 [ Dividing by - 2 and 3 successively ] Method 2 : Steps to find standard form: Find the HCF of numerator and denominator of the fraction. Divide the numerator and denominator by the produced HCF. Write the simplified fraction of the given fraction. The HCF of 18 , 42 is 6 . Thus, we can get its standard form by dividing it by 6 . So, - 18 - 42 = 18 42 ÷ 6 6 = 3 7 Hence, the required answer is as shown above.

Compare 5 17 and - 10 19 .

We have to compare 5 17 and - 10 19 . As we know that, rational numbers are compared taking into consideration the method of comparing both integers and fractions. 5 17 , - 10 19 Denominator must be positive. L.C.M of 17 and 19 = 323 5 17 = 5 17 × 19 19 = 95 323 - 10 19 = - 10 19 × 17 17 = - 170 323 On comparing numerator, we get 95 > - 170 So, we can write 5 17 > - 10 19 . Hence, 5 17 > - 10 19 .

Compare 1 3 and 4 3 .

We have to compare 1 3 and 4 3 . Since the denominators are the same, just compare the numerators. Since 1 < 4 , We conclude that 1 3 < 4 3 . Hence, 1 3 is less than 4 3 .

We have to compare 3 4 and 5 6 . As we know that, rational numbers are compared taking into consideration the method of comparing both integers and fractions. The LCM of the denominators is 12 . Consider for each rational number an equivalent rational number with the LCM 12 as denominator. We get 3 4 = 9 12 and 5 6 = 10 12 , which become like fractions now. Here, 9 12 < 10 12 . Hence, we conclude that 3 4 < 5 6 .

Compare 9 - 4 and - 2 3 .

The number 9 - 4 is not in standard form. First put it in the standard form. 9 - 4 = 9 - 4 × - 1 - 1 to get a positive denominator = - 9 4 Now, we shall compare the fractions - 9 4 and - 2 3 . We find that these two fractions are unlike fractions. To make them as like fractions, we make use of their LCM, which is 12 . We can now compare their equivalent fractions - 9 4 = - 27 12 and - 2 3 = - 8 12 (How?). We find that the denominators are the same and so just comparing the numerators - 27 and - 8 are enough. Visualising these numbers on the number line, we see that - 8 is to the right of - 27 and hence ( - 8 ) > ( - 27 ) . This leads to the result that - 8 12 > - 27 12 and consequently we conclude that - 2 3 > 9 - 4 .

Find a rational number between 1 3 and 5 9 .

We need to find the rational number between 1 3 and 5 9 . The LCM of 3 and 9 is 9 . 1 3 = 1 × 3 3 × 3 = 3 9 , Since, 3 9 < 5 9 , 1 3 < 5 9 A rational number between 1 3 and 5 9 = 1 3 + 5 9 2 = 3 + 5 9 2 = 8 9 2 = 8 9 × 1 2 = 4 9 Hence, a rational between 1 3 and 5 9 is 4 9 .