Let's denote the total number of apples Sara picked as \( x \).
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Calculate the number of defective apples: Sara discarded \( \frac{1}{10} \) of the apples, which amounts to: \[ \text{Defective apples} = \frac{1}{10}x \]
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Calculate the remaining apples: The number of apples remaining after discarding the defective ones is: \[ \text{Remaining apples} = x - \frac{1}{10}x = \frac{9}{10}x \]
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Apples made into cider: It is given that 399 apples were made into cider, which means: \[ \frac{9}{10}x - \text{(apples sold as fresh fruit)} = 399 \]
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Let’s denote the apples sold as fresh fruit as \( y \). Thus, we can write: \[ \frac{9}{10}x - y = 399 \tag{1} \]
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Total remaining apples equation: From equation (1), you can express \( y \) as: \[ y = \frac{9}{10}x - 399 \]
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Substituting into the remaining apples equation:
Since \( y \) is the portion of remaining apples sold as fresh fruit, let's find \( x \):
\[ \frac{9}{10}x = y + 399 \]
Now, substituting \( y = \frac{9}{10}x - 399 \): \[ \frac{9}{10}x - 399 + 399 = \frac{9}{10}x \] This just reconfirms that the algebra holds. To find \( x \), we substitute \( y \):
To find \( x \) we rearranged our equations as follows:
Substituting known values gives: \[ \frac{9}{10}x = 399 + y \]
We know \( y \): \[ y = \frac{9}{10}x - 399 \]
- Now we need to find what fraction of the total apples (which is \( x \)) was sold as fresh fruit \( (y) \): First, we can find \( x \): From \( \frac{9}{10}x - y = 399 \) and knowing \( y \):
We can calculate that
\[
\text{If } y + 399 = \frac{9}{10}x
\]
Because \( y \) itself reduces \( \frac{9}{10}x\).
To actually find the value, let's express \( x \) purely: Solving \( 399 + y \) for when \( y = 0,\) would lead to a calculation based on 399.
We can calculate: \[ \frac{9}{10}x = 399 + y \] Given no further identifiers from y we can use numerical identities from setup 1: \[ \Rightarrow x = 399 \div (9/10)=3990/9= 443.33 \approx valid for testing.
To find fraction, [ \Rightarrow \frac{\frac{9}{10}x - 399}{x} = \frac{\text{apples sold as fresh fruit}}{\text{total}} \]
- Find fresh fruits' fraction from there we seek to represent: If from the previous sum you substitute the terms, You can solve the equations:
Taking the final equation: \[ y = \frac{9x}{10} - 399 \Rightarrow y = \frac{9*443.33}{10}-399 \Rightarrow 80.7 \text{ round down } 80 \]
So from total \( x= 3990 apples\), Final fraction confirming itself we summed returns fraction of total apples: \[ \text{Fraction of apples sold as fresh} = \frac{y}{x} = \frac{80 }{443} =\text{or simplified.}\approx \frac{8}{50} \]
Concluding, \[\textit{Thus, the fraction of the total apples picked that were sold as fresh fruit is } \frac{80}{399}.\] (keep evaluating cross the fractions for summations until clean evaluations).