Super-Hard SAT Math Question 83 | Free SAT Prep | Aniko

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Question 83·200 Super-Hard SAT Math Questions·Algebra

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A 46-pound dog eats two types of dry food: brand R and brand S. The recommended amount of brand R is 1.15 cups for every 16 pounds of body weight per day. The recommended amount of brand S is 0.95 cups for every 14 pounds of body weight per day.

The dog eats a total of 3.2 cups of dry food per day. If $r$ is the number of cups of brand R and $s$ is the number of cups of brand S the dog eats in a given day, and the total amount matches the full recommended amount for a 46-pound dog, what is the value of $r$ ?

When recommendations scale linearly with weight, treat each food’s amount as covering some portion of the dog’s weight. Convert “cups per pounds” to “pounds per cup” (reciprocal), write a linear equation for the total 46 pounds, and combine it with the given total-cups equation. Then substitute to reduce to one variable and solve cleanly, using fractions for the decimals to avoid rounding errors.

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Turn each rate into something you can multiply by cups

Your variables $r$ and $s$ are cups, so use a rate in pounds per cup (take the reciprocal of cups per pound).

You need two equations

One equation comes from “the food amounts must cover 46 pounds,” and the other comes from “the total cups is 3.2.”

Substitute using the total-cups equation

Use $r+s=3.2$ to write $s=3.2-r$ , then plug into the other equation to get a one-variable linear equation.

Desmos Guide

Enter the system in Desmos

Let $x=r$ and $y=s$ . Enter

\frac{16}{1.15}x+\frac{14}{0.95}y=46

and

x+y=3.2.

Find the intersection

Use Desmos’s intersection feature (or visually locate the crossing point). The $x$ -coordinate of the intersection is $r$ .

Read off $r$

Confirm that the intersection’s $x$ -value matches one of the answer choices, and select that choice.

Step-by-step Explanation

Convert each recommendation to pounds per cup

Brand R: $1.15$ cups per $16$ lb means $\frac{1.15}{16}$ cups per lb, so one cup corresponds to

\frac{1}{\frac{1.15}{16}}=\frac{16}{1.15}

pounds.

Brand S: $0.95$ cups per $14$ lb means one cup corresponds to

\frac{1}{\frac{0.95}{14}}=\frac{14}{0.95}

pounds.

Write the two equations

If the dog eats $r$ cups of R and $s$ cups of S, the “weight covered” adds to 46 lb:

\frac{16}{1.15}r+\frac{14}{0.95}s=46.

Also, the total cups is 3.2:

r+s=3.2.

Substitute and solve for $r$

From $r+s=3.2$ , write $s=3.2-r$ and substitute:

\frac{16}{1.15}r+\frac{14}{0.95}(3.2-r)=46.

Use exact fractions $1.15=\frac{23}{20}$ and $0.95=\frac{19}{20}$ , so

\frac{16}{1.15}=\frac{320}{23},\qquad \frac{14}{0.95}=\frac{280}{19}.

Then

\left(\frac{320}{23}-\frac{280}{19}\right)r = 46-3.2\cdot\frac{280}{19}.

Compute each side:

\frac{320}{23}-\frac{280}{19}=\frac{-360}{437}, \qquad 46-3.2\cdot\frac{280}{19}=46-\frac{896}{19}=-\frac{22}{19}.

r=\frac{-\frac{22}{19}}{-\frac{360}{437}}=\frac{22}{19}\cdot\frac{437}{360}=\frac{11\cdot 23}{180}=\frac{253}{180}.

Therefore, the correct choice is $\frac{253}{180}$ .

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation