Maths shelf · Class 7

Dadi's chai for six

Dadi's chai for 4 cups simmers for 5 minutes. For 6 cups, everything goes up by half. Does the simmering time go up to 7½ minutes?

About 35 minutes
Class 7
Mathematics and money

What you will learn: Scale a recipe using ratio and the unitary method, and decide which quantities are proportional and which are not.

What should I know first?

If this feels new, try these first. It is fine to go back.

Which pack is the better buy? Class 5
Fair shares: fractions with roti Class 5

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: Practise · Check yourself · For grown-ups · For teachers

Safety first

Do the maths on paper. Only a grown-up boils anything.

Keep it safe:

Do the scaling on paper. Only a grown-up works at the stove or pours anything hot.
If you use spoons and cups to act it out with cold water, keep small spoons and cardamom pods away from babies: nothing goes in the mouth.

You will need

Paper and a pencilFor the recipe table.
A measuring cup and spoon (optional)To act out the amounts with cold water.
A calculator (optional)To check.

Time: About 35 minutes.

What do you think?

Six cups instead of four: should every number in Dadi's recipe go up by half?

Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.

Yes, every number

Even the simmering time? What does simmering do to the tea? Does more chai need more of it?

Only the amounts

Which numbers are amounts, and which are not? Write your reason for each one.

None of them

Try it. Would 300 mL of water still be enough for 6 cups?

Not sure

That is fine. Take one line of the recipe at a time: if you made only 1 cup, would you need less of it?

Need a hint?

Ask of each number: does each cup need its own share of this?

The story

Dadi makes chai for 4 cups every morning: 300 mL water, 300 mL milk, 4 level teaspoons (tsp) of tea leaves, 4 tsp of sugar and 2 cardamom pods. She has tested it: once it is simmering, 4 cups or 6 cups need about 5 minutes. Today two more customers arrive. "Six cups," says Tara. "So 1½ times everything." Tanmay writes it all down, then stops. "Even the 5 minutes? 7½ minutes of boiling?" Dadi laughs. "Then you would have very strong chai."

The idea

A ratio compares two amounts: water to milk in Dadi's chai is 300 : 300, or 1 : 1.

Two quantities are proportional when one is always the same number of times the other: cups and milk, or tickets and cost.

The unitary method finds the amount for one first, then multiplies: 4 tsp for 4 cups is 1 tsp a cup, so 6 tsp for 6 cups.

These are ingredient amounts for four equal servings in a simple recipe. Some liquid boils away while cooking, so the final amount can change.

Not everything is proportional. Cooking time, temperature, or how long a bus takes do not double when the amount doubles. Always ask why a quantity would grow.

A table of Dadi's chai. For 4 cups: 300 mL water, 300 mL milk, 4 tsp tea, 4 tsp sugar, 2 cardamom pods, simmer 5 minutes. For 1 cup: 75 mL, 75 mL, 1 tsp, 1 tsp, half a pod. For 6 cups: 450 mL, 450 mL, 6 tsp, 6 tsp, 3 pods, simmer about 5 minutes.

Worked example

Milk for 10 cups

A wedding order: 10 cups. How much milk?

For 4 cups: 300 mL. For 1 cup: 300 ÷ 4 = 75 mL.
For 10 cups: 10 × 75 = 750 mL.
Check with a ratio: 10 cups is 2½ times 4 cups, and 2½ × 300 = 750.

750 mL of milk for 10 cups.

Practise

Practise, with help if you need it

Dadi has already dropped 4 cardamom pods into a big pot. How many cups of chai can she make so every cup tastes the same as her 4-cup recipe?

Work it out on paper and write one reason. Then tap the answer closest to yours.

8 cups

Write your reason first. Then check it a second way: 2 pods make 4 cups. What do twice as many pods make?

6 cups

You added 2 cups because 2 more pods went in. Adding is a natural first idea. Check it: open "Show me with cups" and find how many cups one pod makes.

16 cups

That multiplies 4 pods by 4 cups. But the 4 cups go with 2 pods, not with 1. Open "Show me with cups" and find how many cups one pod makes.

Not sure

That is fine. Open "Show me with cups" below, then come back to this question.

Show me with cups

In Dadi's recipe, 4 cups share 2 pods. Put the cups in pairs: 2 pairs, and 1 pod for each pair.

So 1 pod is enough for 2 cups. That is the "one" in the unitary method: here it is one pod, not one cup.

Now count: 4 pods, 2 cups for each pod. How many cups is that?

Four cups of chai in two pairs. Each pair has one cardamom pod above it, labelled "1 pod, 2 cups".

Close this box, look at the question again, and tap your answer. It is fine to change your mind.

Now a new problem, on your own

A painter uses 240 mL of blue paint for 8 equal batches of the same colour. How much blue paint does she need for 5 batches?

Use paper. Decide first whether the answer will be more or less, then work it out.

See a worked answer

First, is the answer more or less than 240 mL? 5 batches are fewer than 8, so less.
Find one batch: 240 ÷ 8 = 30 mL.
Then 5 batches: 5 × 30 = 150 mL.
Check: 150 mL is less than 240 mL, as expected.

150 mL of blue paint for 5 batches.

Can the same calculation tell us how long the paint takes to dry?

See what we think

No. Drying time is not shared out per batch: it depends on the weather, how thick the paint is and the air around it. Like Dadi's simmering time, someone has to test it.

I can change a recipe and explain why the amounts still match. Show a grown-up on paper: how would you convince someone your paint answer is the right size?

Try it

Copy the recipe.
Write Dadi's recipe for 4 cups in a table.
Find one cup.
Divide each amount by 4 to get the amount for one cup.
Scale up.
Multiply each one-cup amount by 6. Check with "1½ times" the 4-cup amounts.
Question every number.
For each line, ask: does each cup need its own share? Mark it "scales" or "does not scale".
Try a strange number.
Scale to 3 cups. What happens to the 2 cardamom pods? What would a cook do in real life?
Make the card.
Make a scaling card with the things that scale and the things that do not, with a reason for each.

What you make (paper or offline)

Your scaling card

A card: water and milk scale, because each cup needs the same; tea and sugar scale, for the same taste per cup; simmering time does not, follow the tested recipe; time to heat up is not simply proportional, it depends on the liquid, pot and flame.

Make a scaling card for a recipe from home, for more people.

Mark what scales and what does not, with a reason.

Ask the cook at home if they agree.

Make it on paper. Keep it with you. Nothing to upload.

What did we find out?

Read this after you have tried it. Look back at your guess and your reason first.

Water, milk, tea, sugar and cardamom are needed per cup, so they are proportional: 6 cups need 1½ times as much.

The unitary method finds one cup first: 300 mL ÷ 4 = 75 mL a cup, so 6 cups need 6 × 75 = 450 mL.

Simmering time is not directly proportional to the number of cups. Dadi's tested recipe uses about 5 minutes for both, so maths alone does not decide it: a cook tests it. How long it takes to reach a simmer depends on the amount of liquid, its starting temperature, the pot and the flame.

Think back

How do you decide whether a quantity will scale?

Why does finding "one cup" first make scaling easier?

Investigate further

Scale the recipe for an office order of 30 cups for grown-ups. Would a cook really use 15 cardamom pods? What else changes with a huge pot? (For a school event, scale a lemonade recipe instead.)
Find two quantities at home that are proportional and two that are not.

Can you use it here?

A bus ticket costs ₹15 a person. Is the cost for 6 people proportional? Is the time the bus takes?

Ask someone at home

Ask the cook at home: when you make food for more people, what do you increase, and what stays the same?

No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.

For grown-upsTap a heading to open it. The safety notes are at the top of the page.

Before you start

Paper and a pencil. If you act it out with spoons and cold water, keep small spoons and cardamom away from babies; nothing in the mouth.
Do any real cooking yourself.

Questions that help

"What is the amount for one cup?"
"Does each cup need its own share of this?"
"Would a cook really do that?"

What's really going on

Children learn "multiply everything" and then apply it to things that do not scale, such as cooking time or temperature.

The unitary method (find one, then multiply) works for any scaling and is easier to check than a ratio in your head.

Real cooks round: 1½ cardamom pods becomes 1 or 2. Talking about this is good maths, not a mistake.

Other ways to do it

Scale a recipe the family really makes.
Use a lemonade recipe instead of chai.

For teachersLearning aims, classroom ideas and what to look for.

Learning aims

Scale quantities using ratio and the unitary method.
Decide whether two quantities are proportional and explain why.
Recognise quantities that do not scale, such as time and temperature in cooking.

Subject links

Maths: ratio and proportion (confirm the chapter in the current edition of your textbook).
Class 5 "Which pack is the better buy?" (price per unit).
Class 9 "A solar system scale model" (scale).

These links are general. They have not been checked against a specific board syllabus.

In the classroom

Groups scale one recipe for a class party and defend each line: "scales" or "does not".
A sorting game: cards with pairs of quantities, sorted into proportional and not.

What to look for

Uses "Show me with cups" if stuck, then solves the paint problem alone, with units. Note on paper whether they needed the help; record nothing on the device.
Finds the one-cup amount first.
Questions whether time and temperature scale.
Rounds sensibly and says why.