Use category percent-share ledgers

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Lesson progressPractice problems 0/5

Difficulty: Advanced
Estimated time: 30 minutes
Domains: Problem Solving and Data Analysis
Techniques: Percent-share-ledgerComplementary-categoriesOverall-percent-changeInverse-share-recoveryMulti-year-percent-change

What you’ll learn

Find the overall percent change when a total is split into categories and each one changes by its own percent.
Set up a share ledger, a small table that follows each category through its change.
Work backward from the overall change to a missing starting share or percent change.
Follow the same categories across more than one year.
Finding the rest of a total: if one part is $30\%$ , the rest is $70\%$ (percent part-whole relationships).
Turning a percent change into a factor: up $20\%$ means multiply by $1.20$ (one-step percent change).
Reading a final factor: $1.06$ means up $6\%$ (reverse and successive percent changes).

Prerequisites

Why this matters on the SAT

Follow every category to the new total

Some SAT questions split a total into categories, give each category its own percent change, and ask how the total changed.

Solution to the example

Imagine the museum starts with $100$ members: $30$ student members and $70$ standard members. Student members go up $20\%$ , which adds $6$ , so there are $36$ . Standard members go down $10\%$ , which removes $7$ , so there are $63$ . That's $36+63=99$ members, down from $100$ . The total fell $1\%$ , so the answer is B.

The table does the same work with decimals, counting the starting total as $1$ instead of $100$ . Each share times its factor is that category's contribution, what it adds to the new total:

Category	Starting share	Category factor	Contribution to new total
Student	$0.30$	$1.20$	$0.30(1.20)=0.36$
Standard	$0.70$	$0.90$	$0.70(0.90)=0.63$

The contributions add to

0.36+0.63=0.99.

That's the $99$ out of $100$ again.

Choice D is the trap: $20\%-10\%=10\%$ . But the $20\%$ acted on only $30$ members and added $6$ , while the $10\%$ acted on $70$ members and took away $7$ . Here, the bigger category loses more members than the smaller one gains. Remember: add the pieces, not the percents.

SAT example

At the start of a year, $30\%$ of a museum's members had student memberships and the rest had standard memberships. During the year, the number of student memberships increased by $20\%$ , while the number of standard memberships decreased by $10\%$ .

What was the percent change in the museum's total membership?

A
$10\%$ decrease
B
$1\%$ decrease
C
$3\%$ increase
D
$10\%$ increase

Example: Combine three category changes

Worked example

In 2025, a company's revenue came $40\%$ from online sales, $35\%$ from store sales, and $25\%$ from partner sales. The next year, online sales increased by $15\%$ , store sales decreased by $10\%$ , and partner sales increased by $8\%$ .

What was the overall percent change in total revenue?

A
$1.5\%$ increase
B
$3.0\%$ increase
C
$4.5\%$ increase
D
$13.0\%$ increase

Step 1

Write the shares and factors

First check that the three shares add up to the entire starting total:

0.40+0.35+0.25=1.

Then turn each change into a factor. Store sales went down, so its factor is below $1$ :

Category	Starting share	Factor
Online	$0.40$	$1.15$
Store	$0.35$	$0.90$
Partner	$0.25$	$1.08$

Step 2

Multiply across each row

Multiply each share by its factor:

Category	Contribution to new total
Online	$0.40(1.15)=0.460$
Store	$0.35(0.90)=0.315$
Partner	$0.25(1.08)=0.270$

Keep each product in its own row, next to its category. If you've left out a share, or used $1.10$ where a $10\%$ decrease needs $0.90$ , it'll stand out.

Step 3

Add the rows and read the result

Add the contributions:

0.460+0.315+0.270=1.045.

Revenue is now $104.5\%$ of what it was, so the increase is

(1.045-1)100\%=4.5\%.

The answer is C.

Calculator loads as you approach

Desmos does the multiplying and adding for this ledger. You can edit the expression to check one of your own.

Check your understanding:

In the example, online sales contribute $0.460$ . Why isn’t that online’s new share of revenue? What does $0.460$ mean?

Work backward, then carry the ledger forward

So far you've known the shares and found the overall change. Harder SAT questions flip that around: they give you the overall change and ask for a starting share or one category's percent change. This is the part most students find hardest. It's the same ledger, with a letter in one spot, and the total row hands you the equation.

Find a missing starting share

With two categories, call one starting share $s$ . The other category has everything else, so its share is $1-s$ .

Say Division A grew by $16\%$ , Division B shrank by $4\%$ , and the total grew by $5\%$ . Here's the ledger:

Category	Starting share	Factor	Contribution
Division A	$s$	$1.16$	$1.16s$
Division B	$1-s$	$0.96$	$0.96(1-s)$
Total	$1$		$1.05$

The contributions add up to the overall factor, $1.05$ :

1.16s+0.96(1-s)=1.05.

Distribute the $0.96$ , collect the $s$ terms, and solve:

\begin{aligned} 1.16s+0.96-0.96s&=1.05\\[1.4em] 0.20s&=0.09\\[1.4em] s&=0.45. \end{aligned}

So Division A started with $45\%$ of the total, and Division B had the other $55\%$ . Check it with $100$ : A's $45$ grows to $52.2$ , B's $55$ shrinks to $52.8$ , and $52.2+52.8=105$ . That's the $5\%$ rise the problem describes.

Check your understanding:

Two departments make up an entire starting budget. Department X grows by $10\%$ , Department Y shrinks by $10\%$ , and the total doesn’t change. What share of the starting budget did each department have?

Common mistake:

It’s easy to think of $s$ and $1-s$ as shares of the new total. They’re shares of the starting total, like the museum’s $30$ and $70$ members out of the first $100$ . Write the starting shares first, then attach the factors.

Follow the categories for more than one year

Two years of change sounds like double the work, but it's the same ledger with one more factor column.

Say Site A and Site B handle all deliveries in Year 1.

From Year 1 to Year 2, Site A grows by $20\%$ and Site B decreases by $5\%$ .
The combined Year 2 total is $5\%$ above the Year 1 total.
From Year 2 to Year 3, Site A decreases by $10\%$ and Site B grows by $20\%$ .

Everything is measured against Year 1, so start by finding the Year 1 split. It's the same backward problem you solved above. Let Site A's share be $s$ :

1.20s+0.95(1-s)=1.05.

Distribute and collect the $s$ terms:

\begin{aligned} 0.25s+0.95&=1.05\\[1.4em] 0.25s&=0.10\\[1.4em] s&=0.40. \end{aligned}

So Site A starts at $0.40$ and Site B at $0.60$ . Out of $100$ Year 1 deliveries, that's $40$ and $60$ .

Now multiply each site's share by its own Year 2 factor and its own Year 3 factor. We want the change through Year 3, so stop after the Year 3 factor:

Category	Year 1 share	Year 2 factor	Year 3 factor	Year 3 contribution
Site A	$0.40$	$1.20$	$0.90$	$0.40(1.20)(0.90)=0.432$
Site B	$0.60$	$0.95$	$1.20$	$0.60(0.95)(1.20)=0.684$

Add the rows to get the overall factor from Year 1 to Year 3:

0.432+0.684=1.116.

So Year 3 is $11.6\%$ above Year 1: every $100$ Year 1 deliveries became $111.6$ .

It's tempting to multiply the Year 2 overall factor, $1.05$ , by a Year 3 factor. But Site A's is $0.90$ and Site B's is $1.20$ , and neither one fits both sites. For example, $1.05(0.90)=0.945$ would say deliveries fell, when they actually rose $11.6\%$ . Keep the two rows separate until the end.

Related: For more practice typing percent equations into Desmos once you've set them up, see Solve percent problems in Desmos.

Every ledger problem has two jobs: the setup and the arithmetic.

Build the ledger by hand when you need to

work out the other starting share, like $62\%$ when one category is $38\%$ ;
match each percent change to its own category;
check which total the answer is compared with, like Year 1’s rather than Year 2’s.

Once the ledger is right, Desmos is handy when

you have several decimal products to add, like $0.40(1.15)+0.35(0.90)+0.25(1.08)$ ;
the equation for a missing share or percent has messy decimals, like $1.16s+0.96(1-s)=1.05$ ;
you need exact values across several years, with no rounding along the way.

Desmos can do the arithmetic and solve the equation. It can’t decide which share and factor belong in each row. That part is yours.

Practice problems

Set up each one by hand, and check first that the starting shares add to $1$ . Before you answer, reread the question and give exactly what it asks for.

Combine two category changes

Practice problem

At the beginning of a season, $45\%$ of a theater's ticket sales were for matinee performances and the rest were for evening performances. The number of matinee tickets sold increased by $12\%$ , while the number of evening tickets sold decreased by $8\%$ .

What was the percent change in the total number of tickets sold?

Calculator loads as you approach

Scratch space for your two share × factor products, if you want it.

Track three program areas

Practice problem

A nonprofit's annual spending is divided among three program areas. Education receives $40\%$ of the spending, health receives $35\%$ , and housing receives $25\%$ .

The next year, education spending increases by $10\%$ , health spending decreases by $20\%$ , and housing spending increases by $8\%$ . By what percent does the nonprofit's total spending decrease?

Your answer

Calculator loads as you approach

If you want, type the sum of your three rows here and compare it with 1.

Recover a starting revenue share

Practice problem

A studio earns all its revenue from subscriptions and licensing. From one year to the next, subscription revenue increases by $18\%$ , licensing revenue decreases by $6\%$ , and total revenue increases by $3\%$ .

What percent of the original total revenue came from subscriptions?

Calculator loads as you approach

Once your equation is written, you can solve it here. Type x in place of s.

Recover one category change

Practice problem

A company's entire production comes from Factory A and Factory B. Factory A originally produced $30\%$ of the total, and Factory B produced the rest. The next year, Factory A's production increased by $15\%$ , while Factory B's production decreased by $p\%$ . Overall production decreased by $1\%$ .

What is the value of $p$ ?

Calculator loads as you approach

You can solve here with x in place of p. Then match the decimal to the exact answer choice.

Carry a recovered split across years

Practice problem

A delivery service operates two hubs, A and B. From 2023 to 2024, Hub A's number of deliveries increased by $20\%$ , Hub B's number decreased by $5\%$ , and the combined number of deliveries increased by $5\%$ .

From 2024 to 2025, Hub A's number of deliveries decreased by $10\%$ , while Hub B's number increased by $20\%$ .

Relative to the combined number of deliveries in 2023, what was the percent increase in the combined number of deliveries in 2025?

Your answer

Calculator loads as you approach

Find the 2023 shares first. Then you can work out each hub’s share times both of its yearly factors here. Don’t round along the way.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A share ledger has one row per category, and the starting shares add to $1$ .
Each row contributes starting share × category factor to the new total. Add the pieces, not the percents.
The sum is the overall factor. Compare it with $1$ : $0.99$ means down $1\%$ .
To work backward, call the starting shares $s$ and $1-s$ , and set the sum of the contributions equal to the known overall factor.
Over several years, keep the first year's shares and multiply each category by its own factor for every year.
Build the ledger by hand. Bring in Desmos only once the shares, the factors, and the total you're comparing with are all right.

Related lesson

When one quantity changes several times in a row, with no categories, review Reverse and combine percent changes.

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

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What you’ll learn

Prerequisites

Follow every category to the new total

Build a share ledger

Example: Combine three category changes

Write the shares and factors

Multiply across each row

Add the rows and read the result

Work backward, then carry the ledger forward

Find a missing starting share

Follow the categories for more than one year

Build the ledger by hand when you need to

Once the ledger is right, Desmos is handy when

Practice problems

Combine two category changes

Track three program areas

Recover a starting revenue share

Recover one category change

Carry a recovered split across years

5 practice examples left

Quick recap

Related lesson

Read distributions and data displays

Practice this lesson