# Order of operations, ratios and multiples

> Read a business formula in the right order, tell a share from a ratio, and turn a multiple into a value.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Numbers and shares
- Lesson 1/36, ~28 min
- Source: https://productovi.com/courses/math-basics/order-of-operations-and-ratios

## Where you work

**Alex Tutor:** Welcome to the course. Most numbers here come from **Klasna**, an online tutoring school from Kyiv that sells prepaid packages of English, exam math and programming lessons. As it grows, more decisions rest on formulas in spreadsheets, and one wrong bracket can quietly cost real money.

**Alex Tutor:** You've just joined Klasna's product team, and your job here is to help with the math behind its numbers, from percentages to probability. Today: the order of operations, shares and ratios, starting with a formula from the finance course for what one lesson earns.

## A formula is a recipe

$$
CM = P - T - 0.025 \times P - V
$$

*CM: contribution margin per lesson; P: price per lesson; T: tutor payout; V: video and AI per lesson; 0.025 × P: a 2.5% payment fee.*

**Sofia:** Lots of letters. Where do I even start?

**Alex Tutor:** Put a number in place of each letter. For an English lesson from a 16-lesson package: P = 620 UAH, T = 390, V = 30. Then one rule: multiplication and division come before addition and subtraction.

**Alex Tutor:** So 0.025 × 620 = 15.50 comes first. Only then subtract, left to right: 620 − 390 − 15.50 − 30 = **184.50 UAH**. That is what one lesson leaves after the costs that come with it.

**Sofia:** What if I just go left to right, the way I read?

**Alex Tutor:** Then you compute (620 − 390 − 0.025) × 620 − 30 and get about 142,555 UAH for one lesson. A result that size should make you stop. Spreadsheets follow the same rule, so a missing bracket gives a wrong number without any warning.

**Alex Tutor:** The full **order of operations**: brackets first, then powers and roots, then × and ÷, then + and −. Within one level, left to right.

![Brackets, powers and roots, × and ÷, then + and −: multiplying first gives 184.50 UAH per lesson, while reading left to right gives an impossible 142,555 UAH.](https://productovi.com/courses/math-basics/diagrams/m1-l1-order-of-operations.png)

![Two cooks follow the same recipe card: one adds ingredients in the right order and gets a neat layered cake, the other gets a collapsed mess.](https://productovi.com/courses/math-basics/inline/order-of-operations-and-ratios-illustration-2.webp)

*Same ingredients, wrong order, a very different result.*

**Alex Tutor:** Try the rule on a bare expression first, with no business meaning to lean on.

> **Exercise:** What is 200 − 40 ÷ 4 × 2?

> **Exercise:** An English lesson from an 8-lesson package costs 650 UAH. The tutor still gets 390, video and AI cost 30, and the payment provider keeps 2.5% of the price. Use CM = P − T − 0.025 × P − V: what is the contribution margin per lesson?

## Shares and ratios

**Sofia:** Then finance divides: CM% = CM ÷ P × 100. What does a **fraction** like that actually tell me?

**Alex Tutor:** Read the fraction line as "per" or "out of". 184.50 ÷ 620 = 0.2976: out of every hryvnia a family pays, about 30 kopiykas stay to cover salaries, the office and profit. Times 100, that is 29.76%.

**Alex Tutor:** That is a **share**: a part divided by the whole it belongs to, so it sits between 0 and 1. A **ratio** divides two different things, such as lessons per learner or LTV ÷ CAC. A ratio can be 7 or 0.3: it says how many times one is the other.

**Sofia:** So the denominator decides what the number means.

**Alex Tutor:** Exactly. 184.50 per lesson and 2,952 per 16-lesson package describe the same lessons with different denominators. Always say "per what".

![The same small pile of blue blocks shown twice: once beside a single lesson card, once beside a whole package box of lesson cards.](https://productovi.com/courses/math-basics/inline/order-of-operations-and-ratios-illustration-3.webp)

*The denominator decides what the number means.*

> **Exercise:** Match each fraction to what it measures.

**Alex Tutor:** Now the denominator. Here's a line from a report draft: what's missing?

> **Exercise:** A report draft says: "An English lesson leaves 2,952 UAH of contribution margin." The margin is 184.50 UAH per lesson, and the package has 16 lessons. What is wrong with the line?

**Sofia:** And in a spreadsheet? There is no fraction line there, just one row of symbols.

**Alex Tutor:** Write it on one line. Everything above the fraction line goes in brackets, because division binds tighter than subtraction. Without them, the sheet divides only the last number.

**Alex Tutor:** Then check the size: a margin can never be more than the price, so CM% must land between 0 and 100. Anything outside that means a bracket is missing.

> **Exercise:** P = 620, T = 390, V = 30, fee 2.5% of the price. Which spreadsheet formula gives CM% correctly?

**Alex Tutor:** And a size check: would you trust this sheet just because it follows the rules?

> **Exercise:** A spreadsheet returns a contribution margin of 142,555 UAH for one 620-UAH lesson. The sheet applies the order of operations correctly, so the result can be trusted.

## Multiples

**Alex Tutor:** One more formula, from the course on building a business without investors. GarageDesk, a subscription app for car repair shops, wants to know what the business is worth.

$$
V \approx \text{ARR} \times m
$$

*V: value of the business; ARR: annual recurring revenue (monthly recurring revenue, MRR, × 12); m: the multiple.*

**Alex Tutor:** m is a **multiple**: a ratio read backwards. Divide a price by ARR and you get m; multiply ARR by m and you get a price. With MRR of 165,000 UAH, ARR is 165,000 × 12 = 1,980,000, and 3× ARR is 5,940,000 UAH.

**Sofia:** And 3× is the same as +300%?

**Alex Tutor:** No: three times as much is +200%, because the original 100% is already inside. More on that next lesson.

![One blue block stands next to a row of three; in the row, the first matches the original and the other two are outlined in orange.](https://productovi.com/courses/math-basics/inline/order-of-operations-and-ratios-illustration-4.webp)

*The original is already inside the three.*

> **Exercise:** A buyer offers 4,000,000 UAH for GarageDesk at closing. GarageDesk’s ARR is 1,980,000 UAH. What multiple of ARR is that? Give two decimals.

**Sofia:** So a multiple always needs its base?

**Alex Tutor:** Always. "Three times revenue" means nothing until you say which revenue: a month’s or a year’s, all of it or only the recurring part. Sellers like the biggest base, buyers the smallest.

**Sofia:** Then I’ll always ask "times what?"

> **Exercise:** GarageDesk’s MRR is 165,000 UAH. A buyer says: "I’ll pay three times your revenue: 3 × 165,000 = 495,000 UAH." This is the same as a 3× ARR offer.

**Alex Tutor:** One last check on the point from earlier: three times as much, in percent.

> **Exercise:** GarageDesk's ARR grew from 660,000 to 1,980,000 UAH. Which description is right?

## Your turn: find the bug

**Alex Tutor:** Spreadsheet mistakes rarely announce themselves. The habit that catches them: know roughly what the answer should be before you look at it.

**Sofia:** Here a lesson can’t leave more than its own price, so anything above 620 is wrong.

**Alex Tutor:** Try it. A colleague typed the contribution margin as `=(620-390-0.025)*620-30` and got a huge number. Explain what went wrong in words that someone without a math background would follow.

> **Exercise:** A colleague typed `=(620-390-0.025)*620-30` to get the contribution margin of a 620-UAH lesson (tutor 390, video and AI 30, fee 2.5% of the price) and got about 142,555 UAH. In 2–4 sentences, explain what went wrong and give the correct formula and result.

## Where you will meet this

> **In short**
>
> - **Order of operations:** brackets, powers and roots, × and ÷, + and −; left to right within a level.
> - **Share:** part ÷ whole, between 0 and 1. **Ratio:** one quantity ÷ another; always say "per what".
> - **Multiple:** value ÷ base, so value ≈ base × m. Three times as much is +200%, not +300%.

> **These lessons use it**
>
> - [Contribution margin: what one lesson leaves](/courses/finance-for-product/contribution-margin): CM and CM% for every Klasna package.
> - [Revenue metrics](/courses/product-analytics/revenue-metrics): ARPU and ARPPU, one numerator with two denominators.
> - [What the business is worth](/courses/bootstrapped-product/what-the-business-is-worth): V ≈ ARR × m and what moves the multiple.

**Alex Tutor:** Next time: percent and percentage points, and why "+1%" can mean two very different things.

- Next lesson: [Percent vs percentage points](https://productovi.com/courses/math-basics/percent-vs-percentage-points.md)

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