use 1,2,3,4,5,6,7 to make fractions smaller than one .How many different fractions smaller than one can be made from these numbers?
To find the number of different fractions smaller than one that can be made using the numbers 1, 2, 3, 4, 5, 6, and 7, we can follow these steps:
1. Identify the numerator: The numerator represents the number on top of the fraction. In this case, we can choose any of the given numbers (1, 2, 3, 4, 5, 6, 7) as the numerator.
2. Identify the denominator: The denominator represents the number on the bottom of the fraction. To create a proper fraction (smaller than one), we need to choose a denominator larger than the numerator. For example, if we choose the numerator as 2, the denominator must be greater than 2 (e.g., 3, 4, 5, 6, or 7).
3. Determine the number of possibilities: To calculate the total number of fractions, we need to count the possible combinations of numerators and denominators.
Let's go through the process step by step:
1. Choose the numerator: We have 7 options (1, 2, 3, 4, 5, 6, 7) to choose from.
2. Choose the denominator: The denominator needs to be greater than the numerator. For example, if we choose 1 as the numerator, the denominator can be any number from the remaining 6 options (2, 3, 4, 5, 6, 7). Similarly, if we choose 2 as the numerator, the denominator can be chosen from the remaining 5 options.
3. Calculate the number of possibilities: We need to add up all the possible combinations for each numerator. Since the denominator options decrease after each choice of numerator, we can sum up the number of possibilities using the following formula:
(number of numerators) + (number of numerators - 1) + (number of numerators - 2) + ... + 1
In this case, the number of numerators is 7. Plugging it into the formula:
7 + 6 + 5 + 4 + 3 + 2 + 1 = 28
Therefore, there are a total of 28 different fractions smaller than one that can be made using the numbers 1, 2, 3, 4, 5, 6, and 7.