Build ratio, rate, and unit chains in Desmos

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Difficulty: Intermediate
Estimated time: 50 minutes
Domains: Problem Solving and Data Analysis Geometry and Trigonometry
Techniques: Expression-chainsDimensional-analysisPart-to-wholeUnit-ratesScale-factor

What you’ll learn

Spot when a ratio, rate, mixture, scale, or unit question needs several connected steps.
Turn a part-to-part ratio into the right fraction of the whole.
Label each in-between value in Desmos so you can check your setup line by line.
Pick conversion factors that cancel the units you start with and leave the units you want.
Square a length scale factor when you convert an area.
Keep every value unrounded until the end.
Check that your final line gives the quantity and unit the question asks for.

Prerequisites

Introduction to Desmos: your SAT calculator0%
Enter fractions, multiplication, parentheses, and exponents in Desmos.

Why this matters on the SAT

Keep a long rate calculation visible

Some SAT ratio and rate questions stack several steps into one problem. If you cram them all into one calculator line, you can't see which rate, time unit, or conversion you used. A short chain of labeled lines keeps every choice in view, so you can check each one.

SAT example

Machine A packages $20$ items in $8$ minutes. Machine B packages $18$ items in $12$ minutes. If both machines work at their constant rates for $1$ hour, how many items do they package in total?

A
$90$
B
$150$
C
$210$
D
$240$

Fast Desmos solution

Give each rate its own line, then the time in minutes, then the total:

a = 20/8
b = 18/12
t = 60
n = (a + b)t

Desmos shows $a=2.5$ items per minute, $b=1.5$ items per minute, and $n=240$ items. The answer is D.

Choice A counts only machine B's output for the hour. Choice B counts only machine A's. Choice C treats machine B's rate as $1$ item per minute.

Notice who did the thinking here. You decided that $a$ and $b$ are rates, that $t$ has to be in minutes, and that the two machines' outputs get added. Desmos only calculated what you typed.

Common mistake:

Expecting Desmos to know what your numbers mean. To Desmos, $60$ is only $60$ . It can’t tell whether a number counts parts, minutes, liters, or grams, or whether it’s a linear scale. Keep a written label for every letter, and check the units yourself.

Calculator loads as you approach

One job per line: two rates, a shared time, then the total number of items.

When should you build a chain?

A chain pays off when the hard part is connecting the steps, not doing any one of them.

Build a labeled Desmos chain when…

The question links two or more rates, ratios, or conversions, like two machines with different rates that run for an hour.
A part-to-part ratio has to become a fraction of the whole mixture, like $2$ parts concentrate to $13$ parts water.
The time is in one unit and the rate uses another, like a rate per minute and a time in hours.
A linear scale for lengths gets used on an area, like $3$ centimeters for every $0.4$ kilometer when you need a region’s area.
There are so many in-between values, like a volume, then a part of it, then a mass, that one long line would be hard to check.

Work it out by hand when…

One quick step finishes the problem, like turning $3$ hours into $180$ minutes or scaling the ratio $2:5$ up to $4:10$ .

Don’t start by typing every number you see. First figure out four things: what the question asks for and in what unit, what you’re starting from, which relationships link the two, and which way each conversion goes.

Check your understanding:

Which one would you build a labeled Desmos chain for: turning $2.4$ liters into milliliters, or finding the mass of one ingredient after you apply a mixture ratio, a flow rate, and an hours-to-minutes conversion? Why?

Make the units tell the story

Think of each rate or conversion as a fraction with units attached. Set up the fraction so the unit you don't want cancels.

Say a testing machine processes $22$ samples every $15$ minutes and runs for $1.5$ hours. You want the number of samples.

The machine's rate is per minute, so first turn the running time into minutes. Hours cancel and leave minutes:

t=1.5\text{ hours}\left(\frac{60\text{ minutes}}{1\text{ hour}}\right)=90\text{ minutes}.

Now apply the sample rate. Minutes cancel and leave samples:

n=90\text{ minutes}\left(\frac{22\text{ samples}}{15\text{ minutes}}\right)=132\text{ samples}.

In Desmos, the chain has the same shape:

t = 1.5(60)
r = 22/15
n = rt

Here's what a written label for each letter looks like:

$t$ is the running time in minutes.
$r$ is samples per minute.
$n$ is the total number of samples.

Try it yourself:

Change the running time from $1.5$ hours to $\frac54$ hour. Will $n$ be more or less than $132$ ? Make your guess, then check the new result.

Common mistake:

Multiplying by $\frac{1}{60}$ because an hour is bigger than a minute. You want a number of minutes, and each hour holds $60$ of them, so multiply by $60$ . A quick size check helps: $1.5$ hours is a lot of minutes, so the number should get bigger, not smaller.

Calculator loads as you approach

The first line gives minutes, the second is samples per minute, and the last gives samples.

Turn a ratio into part of the whole

Say a snack mix has fruit and grain in a ratio of $4:9$ . That does not mean fruit is $\frac49$ of the mix. For every $4$ parts fruit there are $9$ parts grain, so the whole mix has

4+9=13

parts, and fruit is

\frac{4}{4+9}=\frac4{13}

of it. The bottom of the fraction has to count every part of the whole. A short way to remember it: the whole is all the parts added up.

Now say you combine equal weights of two snack mixes:

Mix P has a fruit-to-grain ratio of $4:9$ .
Mix Q has a fruit-to-grain ratio of $3:5$ .

Their fruit fractions are

p=\frac4{4+9} \quad\text{and}\quad q=\frac3{3+5}.

The two mixes weigh the same, so each one counts equally. That makes the fruit fraction of the combination the average of the two:

f=\frac{p+q}{2} =\frac12\left(\frac4{13}+\frac38\right) =\frac{71}{208}.

Check your understanding:

A solution has acid and water in a ratio of $3:17$ . What fraction of the whole solution is acid? What fraction is water?

Common mistake:

Using $\frac4{9}$ as the fruit fraction of Mix P. It’s tempting, because the $4$ and the $9$ sit right there in the ratio. But $\frac49$ compares fruit with grain. Fruit as a part of the whole mix is $\frac4{4+9}$ .

Calculator loads as you approach

Turn each ratio into a fraction of its own whole, then average them because the weights are equal.

Keep rate and time units compatible

A rate only works with a time in the matching unit. If a pump runs at liters per minute, turn the running time into minutes before you multiply. The chain goes

\text{hours} \longrightarrow \text{minutes} \longrightarrow \text{liters}.

Some questions split the time into separate stretches. Use only the stretches the question names. For example, average speed while moving leaves out the time spent stopped, even though the clock kept running.

Before you trust a result, check that the units work out:

\left(\frac{\text{output}}{\text{minute}}\right) \left(\text{minutes}\right) =\text{output}.

If you don't end up with the unit the question asks for, flip a conversion factor or rethink the setup. Changing the arithmetic won't rescue a setup whose units don't work.

Check your understanding:

A pump’s rate is in gallons per second, but its running time is given in minutes. Which time belongs in the final multiplication?

Square a linear scale factor for area

Say $3$ centimeters on a map stands for $0.4$ kilometer in real life. Then the length conversion factor, the linear scale factor, is

\frac{0.4\text{ kilometer}}{3\text{ centimeters}}.

An area has two lengths in it, so both of them need converting. That means using the factor twice, which is the same as squaring it:

\left(\frac{0.4\text{ kilometer}}{3\text{ centimeters}}\right)^2.

Now say a region covers $45$ square centimeters on the map. Since $1$ hectare is $0.01$ square kilometer, the region's actual area is

45 \left(\frac{0.4}{3}\right)^2 \left(\frac{1\text{ hectare}}{0.01\text{ square kilometer}}\right) =80\text{ hectares}.

Desmos will happily calculate 45(0.4/3) if you type it. It doesn't know that an area needs the scale factor twice. That exponent comes from the geometry, so it's up to you.

Try it yourself:

Change the map area from $45$ to $90$ square centimeters and keep the scale the same. Predict the actual area before you read the result.

Common mistake:

Squaring only one number in the scale. Typing 0.4/3^2 squares only the $3$ . Put parentheses around the whole factor, (0.4/3)^2, so the complete factor gets squared.

Calculator loads as you approach

Square the kilometers-per-centimeter factor for area, then turn square kilometers into hectares.

Example: Track every part of a spray mixture

Worked example

A maintenance team prepares a spray using $2$ parts concentrate for every $13$ parts water, by volume. The concentrate contains $180$ grams of active ingredient per liter. The spray is applied at $0.8$ liter per minute for $45$ minutes.

Which choice gives the mass, in grams, of active ingredient applied?

A
$4.8$
B
$864$
C
$5{,}616$
D
$6{,}480$

Step 1

Start from where you’re going

The question wants a mass in grams, so build a path that ends in grams:

\text{minutes} \longrightarrow \text{liters of spray} \longrightarrow \text{liters of concentrate} \longrightarrow \text{grams of active ingredient}.

The problem gives you one relationship for each arrow: the flow rate, the ratio, and the concentration.

Step 2

Turn the ratio into a concentrate fraction

The spray has

2+13=15

parts in all, so concentrate is

f=\frac{2}{2+13}=\frac2{15}

of it. Watch out for $\frac2{13}$ here. That compares concentrate with water, not with the whole spray.

Step 3

Find the total spray and the concentrate in it

First, find how much spray goes out in all:

v=0.8\left(\frac{\text{liter}}{\text{minute}}\right) (45\text{ minutes}) =36\text{ liters}.

Only $\frac2{15}$ of that is concentrate:

c=fv =\frac2{15}(36) =4.8\text{ liters}.

Calculator loads as you approach

Total spray, concentrate, and grams of active ingredient each get their own line, so they can’t get mixed up.

Step 4

Apply the concentration and check the choice

Each liter of concentrate holds $180$ grams of active ingredient, so

g=180\left(\frac{\text{grams}}{\text{liter of concentrate}}\right) (4.8\text{ liters of concentrate}) =864\text{ grams}.

The answer is B.

Choice A stops at $4.8$ liters of concentrate and never gets to grams. Choice C uses the water fraction, $\frac{13}{15}$ . Choice D treats all $36$ liters of spray as concentrate.

Common mistake:

Rounding a line partway through the chain. Keep the original fractions and the full calculator values all the way to the end. If the question wants a rounded answer, round only the final one. You’ll practice this more later, in exactness, rounding, and output verification.

Common mistake:

Reporting the last number without asking what it means. Here, $36$ is the total spray and $4.8$ is the concentrate, but the question asks for the mass of active ingredient. Reread the question’s last sentence, and make sure your final line gives that quantity in that unit.

Finish the solution

The concentrate fraction, the time, and the total volume are already in the calculator. You add the last two lines: liters of concentrate, then grams of active ingredient.

Finish a cleaning-solution chain

Finish the solution

A cleaning solution contains $5$ parts concentrate for every $19$ parts water, by volume. The concentrate contains $150$ grams of active ingredient per liter. The solution is dispensed at $1.2$ liters per minute for $25$ minutes.

What mass, in grams, of active ingredient is dispensed?

First steps

The concentrate is 5/(5 + 19) of the whole solution.
The total volume is the dispensing rate times $25$ minutes.
Add a line for liters of concentrate, then a line for grams of active ingredient.

Finish it

Your answer

Calculator loads as you approach

Finish with c = fv, then g = 150c.

Practice problems

Now it's your turn. Start each problem in a clean calculator, and before you type each line, know what it stands for. Try each problem before you open its hint.

Combine two mixture ratios

Practice problem

Mix P contains dried berries and oats in a weight ratio of $2:7$ . Mix Q contains dried berries and oats in a weight ratio of $3:5$ . Equal weights of the two mixes are combined.

What fraction of the combined weight is dried berries?

Calculator loads as you approach

Turn each ratio into a fraction of its whole, then use the equal weights.

Add rates over a common time

Practice problem

Printer A produces $16$ labels in $12$ minutes. Printer B produces $21$ labels in $18$ minutes. Both printers work at their constant rates for $1$ hour.

How many labels do they produce in total?

Calculator loads as you approach

Put both rates in labels per minute, then use the same 60 minutes for both.

Convert a scaled area

Practice problem

On a uniformly scaled map, $4$ centimeters represents $0.7$ kilometer in reality. A reserve covers $32$ square centimeters on the map.

What is the actual area of the reserve, in hectares?

Use $1$ hectare $=0.01$ square kilometer.

Calculator loads as you approach

Square the whole scale factor, then convert to hectares.

Build the complete ratio and rate chain

Practice problem

A beverage mixture contains $7$ parts extract for every $23$ parts water, by volume. The extract contains $96$ grams of dissolved solids per liter. The mixture flows at $1.25$ liters per minute for $0.4$ hour.

What mass, in grams, of dissolved solids flows during that time?

Your answer

Calculator loads as you approach

Link the extract fraction, the time in minutes, the flow rate, and the concentration.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Build a labeled chain when several ratios, rates, scales, or unit conversions have to work together.
Start from what the question asks for, and plan a unit path from the given facts to that answer.
A part-to-part ratio $a:b$ becomes the part-of-whole fraction $\frac{a}{a+b}$ .
Put the time in the same unit as the rate's "per" unit before you multiply.
For area, square the whole linear scale factor.
Don't round in the middle of the chain.
Desmos does the arithmetic, but you choose every ratio total, rate, conversion direction, and exponent.
Check that your final line gives the exact quantity and unit the question asks for.

Next lesson

Equivalent expressions by graph overlap

Compare algebraic forms by graphing them together and checking that they overlap completely.

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Practice

Practice this lesson

422 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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