Find any two rational numbers between 3/5 and 5/7.

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Points to Remember:

  • Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers, and q is not zero.
  • To find rational numbers between two given rational numbers, we can find their average or use equivalent fractions with a common denominator.

Introduction:

This question requires a factual and analytical approach. We need to find two rational numbers that lie between 3/5 and 5/7. Rational numbers are numbers that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Understanding the concept of rational numbers and their representation is crucial to solving this problem.

Body:

Method 1: Finding the Average

One straightforward method is to find the average of the two given rational numbers. The average of two numbers is their sum divided by two.

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  • Let’s find the average of 3/5 and 5/7: (3/5 + 5/7) / 2

  • To add the fractions, we need a common denominator, which is 35: (21/35 + 25/35) / 2 = (46/35) / 2 = 23/35

  • This gives us one rational number between 3/5 and 5/7.

  • Now, let’s find the average of 3/5 and 23/35: (3/5 + 23/35) / 2 = (21/35 + 23/35) / 2 = (44/35) / 2 = 22/35

  • Therefore, 23/35 and 22/35 are two rational numbers between 3/5 and 5/7.

Method 2: Using Equivalent Fractions

Another approach involves finding equivalent fractions with a larger common denominator.

  • Convert 3/5 and 5/7 to fractions with a common denominator. A common denominator is 35.

  • 3/5 = 21/35

  • 5/7 = 25/35

  • We can see that 22/35 and 23/35 lie between 21/35 and 25/35.

  • Therefore, 22/35 and 23/35 are two rational numbers between 3/5 and 5/7.

Verification:

We can verify our results by converting the fractions to decimals:

Clearly, 0.629 and 0.657 lie between 0.6 and 0.714.

Conclusion:

In conclusion, we have successfully identified two rational numbers, 22/35 and 23/35, that lie between 3/5 and 5/7 using two different methods: finding the average and using equivalent fractions with a common denominator. Both methods demonstrate a clear understanding of rational numbers and their manipulation. This problem highlights the density of rational numbers – infinitely many rational numbers exist between any two distinct rational numbers. There is no single “correct” answer as many rational numbers fulfill the requirement. The methods presented offer a systematic approach to finding such numbers. This exercise reinforces fundamental mathematical concepts and problem-solving skills.

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