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Which of the following is the greatest rational number between the rational numbers 37 and 913?
A) 411
B) 712
C) 12
D) 58

Answer
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Hint: We are given two rational numbers and we need to find out the greatest rational number between them. For that, we will Take LCM of all the rational numbers and after that compare its numerator. The rational number with a greater numerator will be the greatest rational number.

Complete step by step solution:
In this question, we are given two rational numbers and 4 more rational numbers between them and we need to find out which one of these rational numbers is the greatest one.
The given rational numbers are: 37 and 913
And the rational numbers between them are: (1) 411 (2) 712 (3) 12 (4) 58
To check which rational number is greater between the given two numbers, if the denominator of the numbers is not same, then we need to take the LCM and then compare the numerator.
So, the numbers are 411, 712 , 12and 58.
So, we need to find the LCM of 11, 12, 2 and 8.
The LCM of 11, 12, 2 and 8 is 264.
Therefore, multiply and divide 411 with 24, 712 with 22 , 12 with 132 and 58 with 33, we get
411×2424=96264
712×2222=154264
12×132132=132264
58×3333=165264
Now, we have rational numbers with same denominator: 96264, 154264, 132264 and 165264.
So, here we can compare the numerator now. Therefore, here 165 is greater than 96, 154 and 132.
 Hence, 165264 is the greatest rational number between 37 and 913.
Therefore, we can say that 58 is the greatest rational number between 37 and 913.

Note:
Here, we can also check directly which one is greatest by dividing the given numbers. Therefore, we get
411=0.36
712=0.58
12=0.50
58=0.62
Here, 0.62 is greater than 0.36, 0.58 and 0.50. Hence, we can say that 58 is the greatest rational number between 37 and 913.

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