One line of maths, one answer

Here is a tiny puzzle. Work it out in your head before you read on:

2 + 3 × 4 = ?

Many people say 20. They read from left to right like a sentence: 2 + 3 is 5, and 5 × 4 is 20.

Other people say 14. They do the multiplying first: 3 × 4 is 12, and 2 + 12 is 14.

Same numbers. Same signs. Two different answers. That cannot be right! If a shopkeeper and a customer read the same bill and got different totals, there would be an argument. If two engineers read the same instruction and got different numbers, a bridge might not fit together.

Mathematicians all over the world have agreed on one way to read a line like this, so that everyone gets the same answer. The agreed answer is 14. This lesson is about that agreement: what it says, why it makes sense and how to use it without getting tricked.

Riya goes to the kirana shop near her house. She buys one pencil for ₹2 and three erasers at ₹4 each.

How much does she pay? Think about it the way the shopkeeper does:

• The pencil costs ₹2.
• Three erasers at ₹4 each cost 3 × 4 = ₹12.
• Altogether: 2 + 12 = ₹14.

Now write the bill as one line of maths: 2 + 3 × 4. The real answer, the money that actually changes hands, is ₹14. Nobody would pay ₹20! The ₹20 answer would mean Riya bought three lots of (a pencil and an eraser), which is not what happened.

So the rule "multiply before you add" is not a random trick. It matches what the numbers mean. The 3 × 4 is one chunk (the cost of the erasers). It has to be worked out before it can be added to anything else.

Imagine a city where some drivers keep to the left and others keep to the right. Every junction would be a crash waiting to happen. In India we all agree to drive on the left. There is nothing magic about the left side (many countries drive on the right), but everyone agreeing is what keeps us safe.

The order of operations is the same kind of agreement. There is nothing magic about it, but because every textbook, every scientist, every exam and every well-made calculator follows the same order, a line like 2 + 3 × 4 means exactly one thing everywhere in the world: in Chennai, in Chandigarh, in Japan and in Brazil.

Maths is a language. A language only works if the speaker and the listener share the same rules.

Here is the first part of the agreement:

When there are no brackets, do all the multiplying (×) and dividing (÷) first. Then do the adding (+) and subtracting (−).

Let us try it on a few lines. Watch how each line shrinks, one step at a time, until only one number is left. That last number is the value of the line.

• 10 − 2 × 3 = 10 − 6 = 4
• 6 + 8 ÷ 2 = 6 + 4 = 10
• 5 × 2 + 3 × 4 = 10 + 3 × 4 = 10 + 12 = 22
• 20 − 12 ÷ 4 = 20 − 3 = 17

In the third line there are two chunks (5 × 2 and 3 × 4). Work out both chunks, then add them.

Sometimes we really do want to add first. For example, Riya buys 3 packets, and each packet has 2 pencils and 4 erasers. How many things did she buy?

Each packet has 2 + 4 = 6 things, and there are 3 packets. We need to add first and then multiply. To show that, we put the adding inside brackets:

3 × (2 + 4)

Brackets are like a box with a label saying "work me out first". Anything inside the brackets is finished before it is used for anything else.

• 3 × (2 + 4) = 3 × 6 = 18
• (2 + 3) × 4 = 5 × 4 = 20
• (10 − 2) × 3 = 8 × 3 = 24
• 20 ÷ (2 + 3) = 20 ÷ 5 = 4

Compare the first line with 3 × 2 + 4, which is 6 + 4 = 10. Same numbers, same signs, different answer: the brackets changed the meaning.

× and ÷ are partners: neither one is more important than the other. + and − are partners too.

When only partners are left, just go from left to right, the way you read.

• 10 − 3 + 2 = 7 + 2 = 9
• 8 ÷ 4 × 2 = 2 × 2 = 4
• 20 − 5 − 5 = 15 − 5 = 10
• 24 ÷ 6 × 2 = 4 × 2 = 8

It is tempting to think "+ before −" or "÷ before ×", but that is not the rule. Try doing 10 − 3 + 2 with the + first: 3 + 2 = 5, then 10 − 5 = 5. That is wrong! If you have ₹10, spend ₹3 and then someone gives you ₹2, you have ₹9, not ₹5.

Put the three rules together and you have the whole order of operations. In India it is often remembered with the word DMAS (or its longer cousin BODMAS):

• D is for Division
• M is for Multiplication
• A is for Addition
• S is for Subtraction

BODMAS adds B for Brackets at the start and O, which Indian and British books usually read as Of (as in "half of 10") and other books read as Orders, meaning powers. You will meet "of" properly in the next layer.

The word is a memory helper, and it hides one trap you already know about: D and M are partners, and A and S are partners.

Every time you work out a bill, a score or a share, you are secretly using the order of operations. Turning a story into one line of maths is a skill you will use for the rest of your life, so let us practise it.

The trick is to ask: what are the chunks? Each "how many × how much" is a chunk. Chunks get added (or taken away) at the end. If something must be worked out before it is multiplied or shared, put it in brackets.

Here is a surprise. Type 2 + 3 × 4 = into two different calculators.

• A simple calculator (the cheap kind with big buttons, often used in shops) usually shows 20. It works out each step the moment you press the next sign: 2 + 3 = 5, then 5 × 4 = 20.
• A scientific calculator (the kind used in secondary school) or the calculator app on most phones shows 14, because it waits until you press = and then follows the order of operations.

So a simple calculator does not follow the agreed rule. Is it broken? No: it was designed for adding up a list of prices one after another. To get 14 on a simple calculator, you must press the keys in a helpful order: 3 × 4 = 12, then + 2 = 14.

Now that you know the rules, you can catch mistakes, even ones made by grown-ups. A mistake in the order of operations usually looks like one of these:

1. Adding before multiplying: reading 2 + 3 × 4 like a sentence and getting 20.
2. Ignoring the brackets: working out 3 × (2 + 4) as 3 × 2 + 4 = 10 instead of 18.
3. The letter trap: doing + before − (or ÷ before ×) just because of the order of letters in DMAS.

A good detective does not just say "wrong!". They find the exact step where the mistake happened and fix it from there.

Kitchens are full of order-of-operations maths. Think about what Amma works out when guests are coming:

• Rotis: each of the 3 guests eats 4 rotis, and she makes 5 extra for tomorrow's tiffin. That is 4 × 3 + 5 rotis: 4 × 3 + 5 = 12 + 5 = 17.
• Rice: the recipe says 2 cups of rice feed 4 people. For 12 people she needs 12 ÷ 4 × 2 cups: 12 ÷ 4 × 2 = 3 × 2 = 6. First find how many "lots of 4 people" there are, then use 2 cups for each lot.
• Laddoos: she makes 3 trays of 8 laddoos and 6 get eaten while cooling (it happens!). Left over: 3 × 8 − 6 = 18.

In every case the multiplying makes a chunk first, and the adding or taking away happens at the end.

A cricket scorer is always doing order-of-operations maths. A batter's runs are made of chunks: fours, sixes, and runs taken by running. The team total adds the batters' runs and the extras (wides, no-balls and byes).

Suppose Smriti hits 7 fours and 2 sixes and runs 19 more. Her score is

7 × 4 + 2 × 6 + 19 = 7 × 4 + 2 × 6 + 19 = 28 + 2 × 6 + 19 = 28 + 12 + 19 = 40 + 19 = 59 runs.

Without the order of operations, the scorer would get nonsense: reading left to right, 7 × 4 = 28, 28 + 2 = 30, 30 × 6 = 180… a score nobody made!

The Sharma family is going by train from Nagpur to their grandparents' town. An adult ticket costs ₹120, a child's ticket costs ₹60, and there is a ₹20 booking charge for the whole booking.

There are 2 adults and 3 children. The fare is

2 × 120 + 3 × 60 + 20 = 2 × 120 + 3 × 60 + 20 = 240 + 3 × 60 + 20 = 240 + 180 + 20 = 420 + 20 = 440.

Then they buy 5 cups of chai at ₹10 on the platform. The whole trip costs 2 × 120 + 3 × 60 + 20 + 5 × 10 = ₹490.