Write $\frac{2}{5}, \frac{3}{5}$ and $\frac{5}{10}$ in order from smallest to largest.
Answer (1)
Convert all fractions to a common denominator: 5 2 = 10 4 , 5 3 = 10 6 , and 10 5 remains the same.
Compare the numerators: 4, 5, and 6.
Order the fractions based on their numerators: 10 4 < 10 5 < 10 6 .
Write the original fractions in the correct order: 5 2 , 10 5 , 5 3 .
Explanation
Problem Analysis We need to order the fractions 5 2 , 5 3 , and 10 5 from smallest to largest. To do this, we can convert each fraction to have a common denominator, which will allow us to easily compare the numerators.
Finding a Common Denominator The denominators of the fractions are 5, 5, and 10. The least common multiple (LCM) of 5 and 10 is 10. So, we will convert each fraction to an equivalent fraction with a denominator of 10.
Converting to Equivalent Fractions First, let's convert 5 2 to a fraction with a denominator of 10. To do this, we multiply both the numerator and the denominator by 2: 5 2 = 5 × 2 2 × 2 = 10 4 Next, let's convert 5 3 to a fraction with a denominator of 10. To do this, we multiply both the numerator and the denominator by 2: 5 3 = 5 × 2 3 × 2 = 10 6 The fraction 10 5 already has a denominator of 10, so we don't need to change it.
Comparing the Numerators Now we have the fractions 10 4 , 10 6 , and 10 5 . We can easily compare these fractions by comparing their numerators. The numerators are 4, 6, and 5.
Ordering the Fractions The numerators in order from smallest to largest are 4, 5, and 6. Therefore, the fractions in order from smallest to largest are 10 4 , 10 5 , and 10 6 . Now we convert these back to the original fractions: 5 2 , 10 5 , 5 3 .
Final Answer Therefore, the fractions 5 2 , 5 3 and 10 5 in order from smallest to largest are 5 2 , 10 5 , 5 3 .
Examples
Understanding how to order fractions is useful in many real-life situations. For example, imagine you are baking and a recipe calls for 5 2 cup of flour, 5 3 cup of sugar, and 10 5 cup of butter. To make sure you add the ingredients in the correct order, you need to know which fraction is the smallest, the middle, and the largest. Ordering fractions helps you measure the correct amounts and follow the recipe accurately, ensuring your baked goods turn out perfectly!
