Which is greater? 1 2 of 1 3 or 2 3 of 1 5 .

We know that the operator 'of' represents multiplication. Thus, we have, 1 2 of 1 3 = 1 2 × 1 3 = 1 × 1 2 × 3 [The product of two fractions = Product of the numerators Product of the denominators ] = 1 6 . Now, we have, 2 3 of 1 5 = 2 3 × 1 5 = 2 × 1 3 × 5 [The product of two fractions = Product of the numerators Product of the denominators ] = 2 15 . Now, we have to compare 1 6 and 2 15 . For this, we shall convert both fractions into their equivalent fractions such that the denominators of both of them are equal to their LCM. Then, we shall compare the numerators. The fraction with the greater numerator is greater. The LCM of 6 and 15 is 30 . Thus, we have, 1 6 = 1 × 5 6 × 5 = 5 30 and 2 15 = 2 × 2 15 × 2 = 4 30 . Comparing the numerators, we observe, 5 > 4 . Thus, we have, 5 30 > 4 30 and hence, 1 6 > 2 15 . Therefore, 1 2 of 1 3 is greater than 2 3 of 1 5 .

Which is greater? 3 4 of 4 9 or 5 7 of 7 10

We know that the operator 'of' represents multiplication. Thus, we have, 3 4 of 4 9 = 3 4 × 4 9 = 3 × 4 4 × 9 [The product of two fractions = Product of the numerators Product of the denominators ] 3 × 4 4 × 3 × 3 = 1 3 . Now, we have, 5 7 of 7 10 = 5 7 × 7 10 = 5 × 7 7 × 10 [The product of two fractions = Product of the numerators Product of the denominators ] = 5 × 7 7 × 2 × 5 = 1 2 . Now, we have to compare 1 3 and 1 2 . For this, we shall convert both fractions into their equivalent fractions such that the denominators of both of them are equal to their LCM. Then, we shall compare the numerators. The fraction with the greater numerator is greater. The LCM of 3 and 2 is 6 . Thus, we have, 1 3 = 1 × 2 3 × 2 = 2 6 and 1 2 = 1 × 3 2 × 3 = 3 6 . Comparing the numerators, we observe, 3 > 2 . Thus, we have, 3 6 > 2 6 and hence, 1 2 > 1 3 . Therefore, 5 7 of 7 10 is greater than 3 4 of 4 9 .

Find the product of 6 7 and 5 11 and show that it is less than both these fractions.

We have, 6 7 × 5 11 = 6 × 5 7 × 11 [The product of two fractions = Product of the numerators Product of the denominators ] = 30 77 . Now, we have to compare 6 7 , 5 11 and 30 77 . For this, we shall convert the fractions into their equivalent fractions such that the denominators of all of them are equal to their LCM. Then, we shall compare the numerators. The fraction with the greater numerator is greater. The LCM of 7 , 11 and 77 is 77 . Thus, we have, 6 7 = 6 × 11 7 × 11 = 66 77 , 5 11 = 5 × 7 11 × 7 = 35 77 and 30 77 = 30 77 . Comparing the numerators, we observe, 30 < 35 < 66 . Thus, we have, 30 77 < 35 77 < 66 77 or 30 77 < 5 11 < 6 11 . Therefore, the product is less than both the given fractions.

Find the product of 1 4 and 8 5 and show that it is less than 8 5 but more than 1 4 .

We have, 1 4 × 8 5 = 1 × 8 4 × 5 [The product of two fractions = Product of the numerators Product of the denominators ] = 1 × 2 × 4 4 × 5 = 2 5 . Clearly, 2 5 is 8 5 because the denominators of both fractions remaining the same 5 , the numerator 2 of 2 5 is lesser than the numerator 8 of 8 5 . Now, we have to compare 2 5 , 8 5 and 1 4 . For this, we shall convert the fractions into their equivalent fractions such that the denominators of all of them are equal to their LCM. Then, we shall compare the numerators. The fraction with the greater numerator is greater. The LCM of 5 , 5 and 4 is 20 . Thus, we have, 2 5 = 2 × 4 5 × 4 = 8 20 , 8 5 = 8 × 4 5 × 4 = 32 20 and 1 4 = 1 × 5 4 × 5 = 5 20 . Comparing the numerators, we observe, 5 < 8 < 32 . Thus, we have, 5 20 < 8 20 < 32 20 or 1 4 < 2 5 < 8 5 . Therefore, the product is less than 8 5 , but more than 1 4 .

Which is greater: 5 ÷ 1 9 or 1 5 ?

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction. Thus, we have, 5 ÷ 1 9 = 5 × 9 1 [The reciprocal of a fraction is obtained by interchanging its numerator and denominator] = 5 × 9 1 [To multiply a fraction by a whole number, multiply the numerator of the fraction by the whole number and leave the denominator as it is] = 45 1 = 45 . Now, 45 is a whole number greater than 1 , but 1 5 is a proper fraction and its value if less than 1 . Hence, 45 is greater than 1 5 . Therefore, 5 ÷ 1 9 is greater than 1 5 .