I can't understand this math problem....

Please help me....
------------------
Problem: In a boys' school, 5/8 of the boys play football and 4/7 play rugby. If every boy plays at least one of the two games, find the fraction of the boys who play both

since 5/8+4/7 = 67/56, 11/56 play both

check: with 56 students,
35 play football
32 play rugby
so, 11 play both, since all play something.

Sure! Let's break down the problem step by step.

Step 1: Understand the problem
In a boys' school, we are given that 5/8 of the boys play football and 4/7 play rugby. We are also told that every boy plays at least one of the two games. We need to find the fraction of boys who play both football and rugby.

Step 2: Find the common denominator
To find the fraction of boys who play both football and rugby, we need to work with a common denominator. The least common denominator (LCD) of 8 and 7 is 56.

Step 3: Convert the fractions to have the same denominator
To convert the fractions to have the same denominator, we need to find their equivalent fractions with a denominator of 56.

For football:
5/8 = (5/8) * (7/7) = 35/56

For rugby:
4/7 = (4/7) * (8/8) = 32/56

So, 35/56 of the boys play football and 32/56 play rugby.

Step 4: Find the fraction of boys who play both
To find the fraction of boys who play both football and rugby, we need to find the intersection between the two groups. In other words, we need to find the common part between 35/56 and 32/56.

To do this, we multiply the numerators and denominators of both fractions:
(35/56) * (32/56) = (35 * 32) / (56 * 56) = 1120 / 3136

Therefore, the fraction of boys who play both football and rugby is 1120/3136.

Step 5: Simplify the fraction (if necessary)
If the fraction can be simplified, divide both the numerator and denominator by their greatest common divisor (GCD) to simplify the fraction.

In this case, the GCD of 1120 and 3136 is 112. Divide both numbers by 112 to get the simplified fraction:
1120/3136 = (1120/112) / (3136/112) = 10/28

Therefore, the simplified fraction of boys who play both football and rugby is 10/28.

Step 6 (final step): Check if the answer is in simplest or mixed form
If necessary, further simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

In this case, the GCD of 10 and 28 is 2. Divide both numbers by 2 to get the simplest form of the fraction:
10/28 = (10/2) / (28/2) = 5/14

Therefore, the fraction of boys who play both football and rugby is 5/14.

So, the answer is 5/14.

To find the fraction of boys who play both football and rugby, we can use the concept of finding the intersection of two sets.

1. Start by finding the common denominator for 8 and 7, which is 56.

2. Multiply the fractions by the appropriate factors to make the denominators equal to 56.
- For 5/8: Multiply the numerator and denominator by 7
- For 4/7: Multiply the numerator and denominator by 8

This gives us:
- 35/56 for the fraction of boys who play football
- 32/56 for the fraction of boys who play rugby

3. Next, we need to find the fraction of boys who play both games. This can be done by multiplying the two fractions:
- (35/56) * (32/56) = 1,120/3,136

4. Simplify the fraction. Since the numerator and denominator are divisible by 16, we can divide them both by 16:
- (1,120/16) / (3,136/16) = 70/196

5. Finally, simplify the fraction further by dividing both the numerator and denominator by their greatest common divisor, which is 14:
- (70/14) / (196/14) = 5/14

Therefore, the fraction of boys who play both football and rugby is 5/14.