Sophia has a different bag only containing red grapes and green grapes.
The number of grapes in her bag is different, but 4/9 of the grapes are also red.
She picks out a red grape from her bag and eats it.
3/7 of the remaining grapes in her bag are red. How many of the remaining grapes in her bag are red and how many are green?
Let's represent the total number of grapes in Sophia's bag as x. Then, let's say she has y red grapes and z green grapes. We know that:
y + z = x (because the total number of grapes is the sum of the red and green grapes)
4/9 x = y (because 4/9 of the grapes are red)
Now, let's look at what happens when Sophia eats a red grape. There are now x - 1 grapes left in her bag, and y - 1 of them are red. We also know that 3/7 of the remaining grapes (or x - 1) are red. Putting it all together:
3/7 (x - 1) = y - 1
3/7 (x - 1) = 4/9 x - 1
Multiplying both sides by 63 (the least common multiple of 7 and 9), we get:
27(x - 1) = 28x - 63
27x - 27 = 28x - 63
x = 36
Now that we know there are 36 grapes in Sophia's bag, we can use our earlier equations to find that she has 16 red grapes (4/9 of 36). When she eats one of them, she is left with 35 grapes, and we can solve for the number of remaining red grapes:
3/7 (35) = y - 1
y = 16
So there are still 16 red grapes in her bag, and the remaining 19 grapes must be green. Answer:
16 red grapes and 19 green grapes.
Let's break down the information step by step to find the answer.
Step 1: The bag contains red grapes and green grapes.
Let's assume the number of red grapes is represented by 'r' and the number of green grapes is represented by 'g'.
Step 2: 4/9 of the grapes are red.
This means that out of 'r + g' grapes, 4/9 of them are red. So, we can write the equation:
r = (4/9)(r + g)
Step 3: She picks out a red grape and eats it.
After eating a red grape, there are now 'r - 1' red grapes left in the bag.
Step 4: 3/7 of the remaining grapes are red.
This means that out of the 'r - 1' red grapes and 'g' green grapes left in the bag, 3/7 of them are red. So, we can write the equation:
r - 1 = (3/7)(r - 1 + g)
Step 5: Solve the equations to find the values of 'r' and 'g'.
From equation (2):
r - 1 = (3/7)(r - 1 + g)
Multiply both sides by 7 to get rid of the fraction:
7(r - 1) = 3(r - 1 + g)
7r - 7 = 3r - 3 + 3g
4r - 4 = 3g
From equation (1):
r = (4/9)(r + g)
Multiply both sides by 9 to get rid of the fraction:
9r = 4(r + g)
9r = 4r + 4g
Now, we can use these two equations to find the values of 'r' and 'g'.
Solving the two equations simultaneously:
4r - 4 = 3g ...... (3)
9r = 4r + 4g ..... (4)
Multiply equation (4) by -1 and add it to equation (3):
-(4r - 4) + (9r - 4r) = -3g + 4g
-4r + 4 + 5r = g
r + 4 = g
Substitute this value of 'g' in equation (4):
9r = 4r + 4(r + 4)
9r = 4r + 4r + 16
9r = 8r + 16
r = 16
Substitute this value of 'r' in equation (3):
4(16) - 4 = 3g
64 - 4 = 3g
60 = 3g
g = 20
Therefore, there are 16 red grapes and 20 green grapes remaining in her bag.