Muna and Rita had 360 seashells. Muna gave 4/7 of her seashells to Rita. In return, Rita gave 2/5 of her total number of seashells to Muna. They then had an equal number of seashells each. How many seashells did each of them have at first?
Well, let me shell you a secret: Muna and Rita had a shell of a time with their seashells!
Let's solve this sea-worthy puzzle step by step.
Step 1: Muna gave 4/7 of her seashells to Rita. So, she gave away (4/7) * 360 = 160 seashells.
After this exchange, Muna is left with 360 - 160 = 200 seashells, while Rita now has 160 seashells.
Step 2: Rita gave 2/5 of her total number of seashells to Muna. So, she gave away (2/5) * 160 = 64 seashells.
After this exchange, Rita is left with 160 - 64 = 96 seashells. Muna now has 200 + 64 = 264 seashells.
Step 3: The problem states that at this point, they both have an equal number of seashells.
Since they both have the same number of seashells, let's call it 'x'.
We know that Muna has 264 seashells and Rita has 96 seashells. So, we can set up the equation: 264 = 96 + x.
Simplifying, we find that x = 264 - 96 = 168.
So, at first, Muna had 264 seashells and Rita had 168 seashells.
They sure had a lot of seashells shuffling back and forth like that!
Let's break down the problem step by step.
1. Let's assume Muna originally had x seashells and Rita had y seashells.
2. It is given that Muna gave 4/7 of her seashells to Rita. So, Muna has (1 - 4/7) * x = 3/7 * x seashells left.
3. In return, Rita gave 2/5 of her total number of seashells to Muna. So Rita has (1 - 2/5) * y = 3/5 * y seashells left.
4. It is given that they had an equal number of seashells after the exchange. Therefore,
3/7 * x = 3/5 * y
5. To get rid of the fractions, we can multiply both sides by 35 (7 * 5):
35 * (3/7 * x) = 35 * (3/5 * y)
15x = 21y
6. Now, we know that Muna and Rita had a total of 360 seashells:
x + y = 360
7. Now we have two equations:
15x = 21y
x + y = 360
To solve this system of equations, we can use substitution or elimination method.
Let's use the substitution method:
1. From the second equation, we can solve for x in terms of y:
x = 360 - y
2. Substitute this value of x into the first equation:
15(360 - y) = 21y
5400 - 15y = 21y
3. Simplify and solve for y:
5400 = 36y
y = 5400 / 36
y = 150
4. Now substitute the value of y back into the second equation to find x:
x + 150 = 360
x = 360 - 150
x = 210
Therefore, Muna originally had 210 seashells, and Rita originally had 150 seashells.
To solve this problem, we can set up a system of equations.
Let's suppose that Muna had x seashells initially, and Rita had y seashells initially.
According to the given information, Muna gave 4/7 of her seashells to Rita. This means that Muna had (1 - 4/7)x = (3/7)x seashells left, and Rita received (4/7)x seashells.
Next, Rita gave 2/5 of her total number of seashells to Muna. So Muna received (2/5)(4/7)x = (8/35)x seashells and Rita had (1 - 2/5)y = (3/5)y seashells left.
After the exchange, they had an equal number of seashells each. Therefore, we can set up the equation:
(3/7)x + (8/35)x = (3/5)y + (4/7)x
To simplify the equation, we can multiply all terms by the least common denominator, which is 35:
3x + 8x = 15y + 20x
11x = 15y + 20x
11x - 20x = 15y
-9x = 15y
Dividing both sides of the equation by 15:
-9x/15 = 15y/15
-3x/5 = y
Now we can substitute this value of y in terms of x into one of the original equations to solve for x.
Let's substitute y = -3x/5 into the equation:
(3/7)x + (8/35)x = (3/5)y + (4/7)x
(3/7)x + (8/35)x = (3/5)(-3x/5) + (4/7)x
To simplify and get rid of the fractions, we can multiply the equation by the least common denominator, which is 35:
(15/35)x + (8/35)x = (9/25)(-3x) + (20/35)x
(23/35)x = (-27/35)x + (20/35)x
(23/35)x = (-7/35)x
Adding (7/35)x to both sides:
(23/35)x + (7/35)x = (-7/35)x + (7/35)x
(30/35)x = 0
Dividing both sides by 30/35 (or multiplying by the reciprocal):
(35/30)(30/35)x = 0
x = 0
Therefore, Muna initially had 0 seashells.
Now we can substitute x = 0 into the equation y = -3x/5 to find y:
y = -3(0)/5
y = 0
Therefore, Rita initially had 0 seashells as well.
So, both Muna and Rita initially had 0 seashells each.