Find three rational numbers between $\frac{1}{4}$ and $\frac{1}{5}$.
Answer (1)
Find a common denominator for 4 1 and 5 1 , which is 20, so the fractions become 20 5 and 20 4 .
Multiply the numerator and denominator of both fractions by 4 to create more space between them: 80 16 and 80 20 .
Identify three rational numbers between 80 16 and 80 20 : 80 17 , 80 18 , and 80 19 .
Simplify the fraction 80 18 to 40 9 , so the three rational numbers are 80 17 , 40 9 , 80 19 .
Explanation
Understanding the Problem We are given two rational numbers, 4 1 and 5 1 , and we need to find three rational numbers that lie between them.
Finding a Common Denominator First, let's find a common denominator for 4 1 and 5 1 . The least common multiple (LCM) of 4 and 5 is 20. So, we can rewrite the fractions as: 4 1 = 4 × 5 1 × 5 = 20 5 5 1 = 5 × 4 1 × 4 = 20 4
Creating Space for Rational Numbers Now we have 20 4 and 20 5 . Since we need to find three rational numbers between them, we can multiply the numerators and denominators of both fractions by a number greater than 3, say 4. This will give us more space to find the required rational numbers: 20 4 = 20 × 4 4 × 4 = 80 16 20 5 = 20 × 4 5 × 4 = 80 20
Identifying Three Rational Numbers Now we need to find three rational numbers between 80 16 and 80 20 . We can easily find them by picking three numerators between 16 and 20: 80 17 , 80 18 , 80 19
Simplifying the Fractions We can simplify the fraction 80 18 by dividing both the numerator and the denominator by their greatest common divisor, which is 2: 80 18 = 80 ÷ 2 18 ÷ 2 = 40 9
Final Answer Confirmed So, the three rational numbers between 4 1 and 5 1 are 80 17 , 40 9 , and 80 19 .
Final Answer Therefore, three rational numbers between 4 1 and 5 1 are 80 17 , 40 9 , 80 19 .
Examples
Imagine you're baking a cake and the recipe calls for a certain amount of flour, say between 4 1 and 5 1 of a cup. To be more precise, you might want to use 80 17 , 40 9 , and 80 19 of a cup to experiment with the texture and consistency of the cake. Finding rational numbers between two given fractions allows for finer adjustments in measurements, leading to potentially better results in cooking or baking.
