In an aquarium, the ratio of male to female guppies is . During a population study, a researcher removes female guppies and introduces male guppies. After these changes, the ratio of male to female guppies becomes . How many male guppies were in the aquarium before the study began?

Solution available with step-by-step explanation

SAT Ratios, Rates & Proportions Question 15 | Free SAT Prep | Aniko

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Question 15·Hard·Ratios, Rates, Proportional Relationships, and Units

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In an aquarium, the ratio of male to female guppies is $5:8$ . During a population study, a researcher removes $60$ female guppies and introduces $45$ male guppies. After these changes, the ratio of male to female guppies becomes $2:3$ .

How many male guppies were in the aquarium before the study began?

Your answer

(Express the answer as an integer)

For ratio problems with changes (adding or removing items), first represent the original amounts using a single variable multiplied by the ratio parts (e.g., $5k$ and $8k$ for a $5:8$ ratio). Then carefully adjust these expressions for any additions or subtractions described, and use the new ratio to write an equation, usually as a fraction equal to another fraction. Cross-multiply to solve for the variable, and remember that this variable scales the ratio parts—so you must plug it back in (like $5k$ ) to get the actual quantity the question asks for. Finally, do a quick check by plugging your numbers back into the story to see if the final ratio matches.

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Turn the ratio into algebra

If the ratio of male to female guppies is $5:8$ , how can you represent the number of males and females using a single variable $k$ ?

Account for the changes

After 45 males are added and 60 females are removed, write expressions for the new numbers of males and females in terms of $k$ .

Use the new ratio to form an equation

The new ratio of males to females is $2:3$ . How can you write an equation using $\dfrac{5k+45}{8k-60}$ and $\dfrac{2}{3}$ ?

Solve and interpret

Once you solve for $k$ , remember that $k$ is not the number of fish; you still need to use it to find the original number of male guppies.

Desmos Guide

Set up the equation in Desmos

In Desmos, enter the left side of the ratio as a function of $x$ :

y = (5x + 45) / (8x - 60)

This represents the new male-to-female ratio after the changes.

Graph the target ratio

On a new line, enter the constant ratio:

y = 2/3

This is the ratio the problem says we should have after the changes.

Find the value of k and then the initial males

Look at the point where the two graphs intersect and read the x-coordinate; this is the value of $k$ . Then, on a new line, type 5 * (that x-value) to have Desmos calculate the initial number of male guppies.

Step-by-step Explanation

Represent the initial numbers with a variable

The initial ratio of male to female guppies is $5:8$ .

This means we can write:

Number of male guppies $=5k$
Number of female guppies $=8k$

for some positive number $k$ .

Write expressions after the changes

During the study:

45 male guppies are added, so new males $=5k+45$ .
60 female guppies are removed, so new females $=8k-60$ .

After these changes, the ratio of male to female guppies becomes $2:3$ , so:

\frac{5k+45}{8k-60} = \frac{2}{3}

Solve the ratio equation for k

Cross-multiply to solve for $k$ :

3(5k+45) = 2(8k-60)

Simplify both sides:

15k + 135 = 16k - 120

Bring like terms together:

\begin{aligned} 135 + 120 &= 16k - 15k \\ 255 &= k \end{aligned}

So $k=255$ .

Find the initial number of male guppies

The initial number of male guppies is $5k$ .

Substitute $k=255$ :

5k = 5 \times 255 = 1275

So, there were 1275 male guppies in the aquarium before the study began.

Strategy

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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