3 apples and a pineapple equally balances 10 oranges. Also,6 oranges and an apple equally balances a pineapple. How many oranges balance a pineapple?
3 a + p = 10 o
6 o + a = p write as 3 a + 18 o = 3 p
so now
3 a + p = 10 o
3 a + 18 o = 3 p
------------------subtract
p - 18 o = 10 o - 3 p
4 p = 8 0
1 p = 2 o
so
two oranges balance a pineapple
Let's assume the number of oranges required to balance a pineapple is 'x'.
According to the first scenario, 3 apples and a pineapple equally balance 10 oranges.
This can be expressed as (3 apples + 1 pineapple) = 10 oranges.
Now, let's consider the second scenario, where 6 oranges and an apple equally balance a pineapple.
This can be expressed as (1 apple + 6 oranges) = 1 pineapple.
Since both scenarios involve balancing a pineapple, we can equate the two expressions:
(3 apples + 1 pineapple) = (1 apple + 6 oranges)
By rearranging the terms, we get:
(3 apples - 1 apple) = (6 oranges - 1 pineapple)
Simplifying further, it becomes:
2 apples = 6 oranges - 1 pineapple
Now, let's substitute the value of apples in terms of oranges from the first scenario:
2(10 oranges - 1 pineapple) = 6 oranges - 1 pineapple
Expanding and rearranging the equation, we get:
20 oranges - 2 pineapples = 6 oranges - 1 pineapple
Combining like terms, we have:
20 oranges - 6 oranges = 1 pineapple - 2 pineapples
14 oranges = -1 pineapple
Since the number of oranges cannot be negative, this implies that there is no solution to this system of equations. Hence, we cannot determine the exact number of oranges that balance a pineapple based on the given information.
To solve this problem, we need to set up a system of equations. Let's assume that "x" represents the number of oranges needed to balance a pineapple.
From the first statement, we know that 3 apples and a pineapple equally balance 10 oranges. This can be written as:
3 apples + 1 pineapple = 10 oranges
From the second statement, we know that 6 oranges and an apple equally balance a pineapple. This can be written as:
6 oranges + 1 apple = 1 pineapple
Now we have a system of two equations:
Equation 1: 3 apples + 1 pineapple = 10 oranges
Equation 2: 6 oranges + 1 apple = 1 pineapple
To solve this system, we can use substitution.
From Equation 2, we can solve for the number of apples:
1 apple = 1 pineapple - 6 oranges
Now we substitute this value into Equation 1:
3(1 pineapple - 6 oranges) + 1 pineapple = 10 oranges
Expanding the equation:
3 pineapples - 18 oranges + 1 pineapple = 10 oranges
Combining like terms:
4 pineapples - 18 oranges = 10 oranges
Moving all the terms with oranges to one side:
4 pineapples = 28 oranges
Dividing both sides by 4:
pineapples = 7 oranges
Therefore, a pineapple balances 7 oranges.