My son had the following word problem and I don't know where to begin:The ratio in the school gym of boys to girls is 4:3. 160 boys left the gym and the ratio became 4:5. How many girls are in the gym?
4/3 = b/g so b = 4 g/3
4/5 = (b-160)/g
or
4 g = 5 (4g/3 - 160)
4 g = 20 g/3 - 800
12 g = 20 g - 2400
so
8 g = 2400
g = 300
To solve this word problem, we need to use the concept of ratios and proportions. Let's break it down step by step:
Step 1: Understand the problem.
The problem states that the ratio of boys to girls in the gym is 4:3. This means that for every 4 boys, there are 3 girls.
Step 2: Identify the given information.
The problem also states that 160 boys left the gym and the ratio became 4:5. This information tells us that the new ratio of boys to girls is 4:5.
Step 3: Set up a proportion.
We can set up a proportion to find the number of girls in the gym. Let's assign variables to the unknowns:
Let x be the initial number of girls in the gym.
The initial ratio: boys to girls = 4:3
After 160 boys left: boys to girls = 4:5
We can represent these ratios as fractions:
Initial ratio: (boys / girls) = 4/3
After 160 boys left: (boys / girls) = 4/5
We can set up a proportion by equating the two ratios:
(4/3) = (4/5)
Step 4: Solve the proportion.
To solve the proportion, we can cross-multiply.
3 * 4 = 5 * 4
12 = 20
Since 12 is not equal to 20, our assumption that x represents the initial number of girls in the gym is incorrect. We need to introduce another variable.
Let y represent the number of girls left in the gym after 160 boys left.
Now, we can set up a new proportion:
(boys / girls) = (4/3)
(boys - 160) / y = (4/5)
Step 5: Solve the new proportion.
Cross-multiply:
5 * (boys - 160) = 4 * y
5 * boys - 800 = 4 * y
Step 6: Substitute values and solve for y.
We know that the initial number of boys is 4 times the initial number of girls, so we can substitute:
boys = 4x
Substituting, we get:
5 * (4x - 160) = 4 * y
20x - 800 = 4 * y
20x - 800 = 4y
Simplifying the equation:
4y = 20x - 800
y = (20x - 800) / 4
y = 5x - 200
Step 7: Find the value of x.
Now, we need to find the value of x, which represents the initial number of girls in the gym.
Since the ratio of boys to girls is initially 4 to 3, we know that:
4x = 3y
Substituting the value for y:
4x = 3(5x - 200)
4x = 15x - 600
-11x = -600
x = (-600) / (-11)
x ≈ 54.55
Therefore, the initial number of girls in the gym is approximately 54.55.
Note: Since it's not possible to have a fractional or decimal number of girls, we can either round down to 54 or round up to 55. It's up to you to decide which option makes more sense in the context of the problem.
Thus, there are either 54 or 55 girls in the gym initially.