The table shows the average price per kilogram of three varieties of organic produce at a farmers market. | Produce | Price (USD per kg) | | --- | --- | | Apples | 3.50 | | Carrots | 2.25 | | Kale | 4.75 | If a shopper spends exactly \ackkac$?

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SAT Linear Functions Question 48 | Free SAT Prep | Aniko

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Question 48·Hard·Linear Functions

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The table shows the average price per kilogram of three varieties of organic produce at a farmers market.

Produce	Price (USD per kg)
Apples	3.50
Carrots	2.25
Kale	4.75

If a shopper spends exactly $100 on $a$ kilograms of apples, $c$ kilograms of carrots, and $k$ kilograms of kale, which of the following expresses $k$ in terms of $a$ and $c$ ?

For questions that ask you to express one variable in terms of others from a word problem, first translate the situation into an equation using the table or description, then clear decimals or fractions to make the arithmetic easier. Next, carefully isolate the requested variable with inverse operations (add/subtract, then divide) and simplify the resulting fraction by factoring out common divisors. Finally, match your simplified expression to the answer choices, watching for sign errors or swapped coefficients that often create tempting wrong options.

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Write the total cost equation

Use the prices in the table to write an equation for the total cost of apples, carrots, and kale, and set it equal to $100$ .

Avoid working with decimals

Once you have the equation, consider multiplying everything by $100$ so that all numbers are whole (no decimals).

Isolate and solve for $k$

After clearing decimals, move the terms with $a$ and $c$ to the other side of the equation, then divide by the coefficient of $k$ and simplify the fraction as much as possible.

Desmos Guide

Enter the original cost equation

In Desmos, type 3.5*a + 2.25*c + 4.75*k = 100. This represents the relationship among $a$ , $c$ , and $k$ from the problem.

Test each answer choice algebraically

For each option, solve the equation in Desmos by replacing $k$ with the expression from that option. For example, for one choice type 3.5*a + 2.25*c + 4.75*((400 - 14*a - 9*c)/19) and let Desmos simplify. The correct expression will make the left-hand side simplify exactly to 100 (for all $a$ and $c$ ), while incorrect ones will not.

Verify using sample values

Alternatively, pick simple values (like $a=1$ , $c=1$ ) by creating sliders for a and c. For each answer choice, use its expression for $k$ in Desmos, compute the total cost $3.5a + 2.25c + 4.75k$ , and see which choice gives a total of exactly $100$ ; that one corresponds to the correct formula for $k$ .

Step-by-step Explanation

Translate the situation into an equation

Each type of produce has a price per kilogram.

Apples: $3.50$ dollars per kg, so cost is $3.50a$ .
Carrots: $2.25$ dollars per kg, so cost is $2.25c$ .
Kale: $4.75$ dollars per kg, so cost is $4.75k$ .

The total cost is $100$ dollars, so the equation is:

3.50a + 2.25c + 4.75k = 100.

Clear the decimals to make solving easier

Multiply every term in the equation by $100$ to work in cents instead of dollars:

100(3.50a) + 100(2.25c) + 100(4.75k) = 100(100).

This gives:

350a + 225c + 475k = 10000.

Isolate the term with $k$

Subtract $350a$ and $225c$ from both sides so that the $k$ term is alone on the left:

475k = 10000 - 350a - 225c.

Solve for $k$ and simplify the fraction

Divide both sides of the equation by $475$ :

k = \frac{10000 - 350a - 225c}{475}.

Now divide the numerator and denominator by $25$ (their greatest common factor):

$10000 \div 25 = 400$
$350 \div 25 = 14$
$225 \div 25 = 9$
$475 \div 25 = 19$

So the simplified expression for $k$ is

k = \frac{400 - 14a - 9c}{19},

which matches choice D.

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Step-by-step Explanation

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Step-by-step Explanation