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प्रश्न
Insert 9 rational numbers between the following rational numbers:
`3/10` and `7/5`
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उत्तर
Given: Insert 9 rational numbers between the rational numbers`3/10` and `7/5`.
Stepwise calculation:
1. Let the two rational numbers be `a = 3/10, b = 7/5`.
2. Convert to equivalent fractions with a common denominator or directly calculate the difference:
`b - a = 7/5 - 3/10`
= `14/10 - 3/10`
= `11/10`
3. We need to insert n = 9 rational numbers between a and b.
Using the formula for numbers between two rational numbers:
Step size = `(b - a)/(n + 1)`
= `(11//10)/10`
= `11/100`
4. The (k)-th rational number between a and b is:
a + k × Step size = `3/10 + k xx 11/100`
5. Calculating each number for k = 1, 2,...,9:
`k = 1 : 3/10 + 11/100`
= `30/100 + 11/100`
= `41/100`
`k = 2 : 3/10 + 2 xx 11/100`
= `30/100 + 22/100`
= `52/100`
= `13/25`
`k = 3 : 3/10 + 3 xx 11/100`
= `30/100 + 33/100`
= `63/100`
`k = 4 : 3/10 + 4 xx 11/100`
= `30/100 + 44/100`
= `74/100`
= `37/50`
`k = 5 : 3/10 + 5 xx 11/100`
= `30/100 + 55/100`
= `85/100`
= `17/20`
`k = 6 : 3/10 + 6 xx 11/100`
= `30/100 + 66/100`
= `96/100`
= `24/25`
`k = 7 : 3/10 + 7 xx 11/100`
= `30/100 + 77/100`
= `107/100`
`k = 8 : 3/10 + 8 xx 11/100`
= `30/100 + 88/100`
= `118/100`
= `59/50`
`k = 9 : 3/10 + 9 xx 11/100`
= `30/100 + 99/100`
= `129/100`
The 9 rational numbers between `3/10` and `7/5` are:
`41/100, 13/25, 63/100, 37/50, 17/20, 24/25, 107/100, 59/50, 129/100`
Alternatively, a similar set of 9 rational numbers provided is:
`46/75, 77/75, 36/25, 139/75, 34/15, 67/25, 232/75, 263/75, 98/25`
Or, simplified within the interval, as from the source:
`2/5, 1/2, 3/5, 7/10, 4/5, 9/10, 1, 11/10, 6/5`
The calculation method is by equally dividing the interval between the two rational numbers into 10 parts 9 rational numbers inserted.
The methods are from the provided source, detailing the exact rational numbers between `3/10` and `7/5`.
