Sophia buys an eqaul number of oranges and pears for a party. The oranges are bought at a price of 7 for $2 and the pears are bought at a price of 5 for $3. She pays $33 more for the pears than the oranges. How much does Sophia pay in all? How many oranges and pears does she buy altogether?
yup
X = # of oranges.
X = # of pears.
($2/7) * X = $Y
Multiply both sides by 7:
2x = 7y
Eq1: 2x - 7y = 0
($3/5) * X = y + 33
Multiply both sides by 5:
3x = 5y + 165
Eq2: 3x - 5y = 165
Multiply Eq1 by 3 and Eq2 by -2 and add:
+6x - 21y = 0
-6x + 10y = -330
Sum: -11y = -330
Y = $30 = Cost of oranges
Y+33 = 30 + 33 = $63 = Cost of pears
Total cost = 30+63 = $93.
In Eq1, replace y with 30:
2x - 7*30 = 0
2x = 210
X = 105 of each fruit.
Tot. fruit = 2*105 = 210.
To solve this problem, let's first set up some equations. Let's assume that Sophia buys x oranges and x pears.
Given that the price of 7 oranges is $2, we can determine the price of a single orange by dividing $2 by 7. So the cost of one orange is $2/7.
Similarly, given that the price of 5 pears is $3, we can determine the price of a single pear by dividing $3 by 5. So the cost of one pear is $3/5.
According to the problem, Sophia pays $33 more for the pears than the oranges. This can be represented as:
(x pears cost) = (x oranges cost) + $33.
Since the cost of x oranges is x times the cost of one orange, we can write:
(x * $2/7).
Similarly, the cost of x pears can be written as:
(x * $3/5).
Putting it all together, we have the equation:
(x * $3/5) = (x * $2/7) + $33.
To solve this equation, we can start by multiplying both sides of the equation by 35 (a common multiple of 5 and 7) to eliminate the denominators:
35 * (x * $3/5) = 35 * (x * $2/7) + 35 * $33.
This simplifies to:
21x = 10x + $1155.
Next, we can subtract 10x from both sides of the equation:
21x - 10x = $1155.
This further simplifies to:
11x = $1155.
Now, divide both sides of the equation by 11 to solve for x:
x = $1155 / 11.
Calculating this division, we find that x = 105. Therefore, Sophia buys 105 oranges and 105 pears.
To find out how much Sophia pays in total, we need to calculate the cost of the oranges and the pears separately.
The cost of the oranges is given by:
105 * cost of one orange = 105 * ($2/7).
Calculating this, we find that the cost of oranges is $30.
The cost of the pears is given by:
105 * cost of one pear = 105 * ($3/5).
Calculating this, we find that the cost of pears is $63.
In total, Sophia pays $30 + $63 = $93.
Therefore, Sophia pays a total of $93 for buying 105 oranges and 105 pears altogether.