Percentages, proportion and growth. Attnom Academy

M4.2.1 Chapter 1

Percentage increase and decrease

How do I add 12% to 250 in one step?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Add 12 to 250 to get 262.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Multiply 250 by 1.12 to get 280.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Multiply 250 by 0.12 to get 30.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Increase 250 by 12% using the multiplier 1.12
Find the percentage change from 80 to 92

A packet holds 250 seeds. A new packet holds 12% more. How many seeds is that?

Percent means out of 100. So 12% is 12/100, which is 0.12. 12% of 250 is 0.12 × 250 = 30. Add that to the start: 250 + 30 = 280.

That took two steps. There is a way to do it in one.

Draw it first

Add the rise to the start

1 The start is 100% of the seeds. That bar is 250 seeds.
2 The rise adds 12% more. The new bar is 112% of the start, and the extra piece is dotted.
3 112% is 1.12 as a decimal. One multiplication takes 250 to 280.

The start is 100%. A rise of 12% makes 100% + 12% = 112%. As a decimal, 112% is 1.12. The number 1.12 is the multiplier. Multiply by it, and the rise is done.

Worked example

Increase 250 by 12% using a multiplier.

A rise of 12% makes 100% + 12% = 112% of the start.
112% as a decimal is 1.12. This is the multiplier.
Multiply the start by the multiplier: 250 × 1.12 = 280.

Answer: 250 × 1.12 = 280

A fall works the same way. A fall of 35% leaves 100% − 35% = 65% of the start. As a decimal that is 0.65. The multiplier for a fall is below 1.

A fall of 35% leaves 65% of the start. The missing 35% is the dashed part.

Worked example

Decrease 240 by 35%.

A fall of 35% leaves 100% − 35% = 65% of the start.
65% as a decimal is 0.65. This is the multiplier.
Multiply: 240 × 0.65 = 156.

Answer: 240 × 0.65 = 156

A rise adds to 1. A fall takes away from 1.

Sometimes you are given the old and the new amount and asked how much it changed. A percentage change says the change as a percent of the amount you started with.

Work out the change: new minus original. Divide it by the original. Multiply by 100. As a formula: percentage change = (new − original) ÷ original × 100.

The dotted piece is the change, 12 seeds. It is measured against the original, which is 100%.

Worked example

Find the percentage change from 80 to 92.

The change is new minus original: 92 − 80 = 12.
Divide the change by the original: 12 ÷ 80 = 0.15.
Multiply by 100: 0.15 × 100 = 15.
The amount rose by 15%.

Answer: A rise of 15%

See it three ways

With things

Take 80 stones. Put them in 10 piles of 8. Add 12 more stones. That is one more pile and half a pile.

As a picture

In symbols

12 ÷ 80 × 100 = 15%.

Try it

Cut a strip

You need: paper, a ruler, a pencil.

Draw a strip 10 cm long. Let the whole strip stand for 100%. Mark every 1 cm.
Shade the first 6.5 cm from the left. Leave the last 3.5 cm white.

What you will see: The shaded part is 65% of the strip. A fall of 35% leaves 65%, so the multiplier is 0.65.

If you cannot try it, this is what happens.

A wrong idea many people have

To increase 250 by 12%, add 12: 250 + 12 = 262.

Why it feels right: The words say "12" and "add", so it feels natural to add the 12 you can see.

The test that breaks it: Work out 12% of 250. It is 0.12 × 250 = 30, not 12. The rise is 30 seeds, so 262 is too low.

What is true: The percent is a fraction of the start, not a number to add. Use the multiplier: 250 × 1.12 = 280.

A wrong idea many people have

The amount falls from 125 to 100, so the fall is 25 out of 100, which is 25%.

Why it feels right: The new amount is the one you are looking at, and 100 makes the sum look like a percent already.

The test that breaks it: A fall of 25% would leave 125 × 0.75 = 93.75, not 100. Check by multiplying.

What is true: Divide by the original amount, 125. The change 25 ÷ 125 = 0.2, so the fall is 20%.

Key words

percent: out of 100. So 12% is 12 out of 100, or 0.12.
multiplier: the number you multiply by to change an amount, such as 1.12 for a rise of 12%.
percentage change: the change divided by the original amount, times 100.

Back to your guess

The answer is B: Multiply 250 by 1.12 to get 280.

12% of 250 is 30, and that is what 0.12 gives. The rise adds 30 to the 250, so the new amount is 280. Multiplying by 1.12 does both steps at once.

Keep it

Copy this into your exercise book.

Rise: multiply by 1 + the rate. Fall: multiply by 1 − the rate. Percentage change = change ÷ original × 100. Always divide by the original.

Draw: A bar of 100% and a longer bar of 112%, with an arrow labelled × 1.12 between them.

M4.2.2 Chapter 2

Finding the original amount

A herd grew by 25% and now has 150 goats. How many goats did it have before?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Take 25% off 150, which leaves 112.5.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Take 25 off 150 to get 125.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Divide 150 by 1.25 to get 120.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Find the original amount after a 20% rise gives 96
Explain why a 10% rise then a 10% fall is a loss

Last year a herd had some goats. It grew by 20% and now has 96 goats. We want the number it started with, the original amount.

The start is 100%. After a rise of 20%, the herd is 120% of the start. So 96 goats is 120% of the original amount.

Draw it first

Work back from the new amount

1 The new herd is 96 goats. That is 120% of the original.
2 The old herd is the shorter bar. We do not know it yet.
3 The rise multiplied the old herd by 1.2. To go back, divide the new herd by 1.2.

A rise multiplies by 1.2. Going back undoes the multiplying, so divide by 1.2. Divide by the multiplier, whether the change was a rise or a fall. To divide by 1.2, multiply both numbers by 10 first: 96 ÷ 1.2 = 960 ÷ 12 = 80.

Worked example

A herd grew by 20% and now has 96 goats. Find the original amount.

The new amount is 100% + 20% = 120% of the original.
120% as a decimal is 1.2. So original × 1.2 = 96.
Go back by dividing by the multiplier: 96 ÷ 1.2 = 80.
Check: 80 × 1.2 = 96. So the herd had 80 goats before.

Answer: 96 ÷ 1.2 = 80

The rise is 20% of the old herd: 16 goats. It is not 20% of the new herd.

A wrong idea many people have

To undo a rise of 20%, take 20% off 96.

Why it feels right: A rise and a fall of the same percent sound like opposites that cancel.

The test that breaks it: 20% of 96 is 19.2, and 96 − 19.2 = 76.8. Now check by rising 20%: 76.8 × 1.2 = 92.16, not 96.

What is true: The rise was 20% of the old amount, 80, and not 20% of 96. Divide by 1.2 to reach 80.

A fall works the same way. The multiplier is below 1, and you still divide by it.

140 litres is 70% of the start.

Worked example

A tank lost 30% of its water and now holds 140 litres. How much did it hold at the start?

A fall of 30% leaves 100% − 30% = 70% of the start.
70% as a decimal is 0.7. So start × 0.7 = 140.
Divide by the multiplier: 140 ÷ 0.7 = 200.
The tank held 200 litres. Check: 200 × 0.7 = 140.

Answer: 140 ÷ 0.7 = 200

Now a puzzle. A herd of 200 goats grows by 10%. Then it falls by 10%. Is it back where it began?

The herd ends at 198, which is 2 below where it began.

It is not back. The rise of 10% was worked out on 200. The fall of 10% was worked out on 220, which is a bigger amount. A bigger amount loses more.

Worked example

A herd of 200 goats grows by 10%, then falls by 10%. How does the end compare with the start?

A rise of 10% has multiplier 1.1: 200 × 1.1 = 220.
A fall of 10% has multiplier 0.9: 220 × 0.9 = 198.
One multiplier does both: 1.1 × 0.9 = 0.99.
The herd ends at 0.99 of the start, which is 1% less.

Answer: Less, by 1%

See it three ways

With things

Take 100 stones. Add 10%: that is 10 more, so 110 stones. Now take away 10% of the 110: that is 11 stones. 99 are left.

As a picture

In symbols

100 × 1.1 × 0.9 = 99.

Try it

Try any number

You need: paper, a pencil.

Write any whole number you like. Here we use 300.
Raise it by 10%. Write the answer.
Lower that answer by 10%. Write the new answer.

What you will see: You end below where you began. With 300 you reach 297. Try other numbers: the end is always 0.99 of the start.

If you cannot try it, this is what happens.

A wrong idea many people have

A rise of 10% and then a fall of 10% cancel, so you end where you began.

Why it feels right: Plus 10 and minus 10 do cancel. The same number appears twice, so it feels the same here.

The test that breaks it: Take 100. A rise of 10% gives 110. A fall of 10% takes off 10% of 110, which is 11. That leaves 99, not 100.

What is true: Each percent is of a different amount. Multiply the multipliers: 1.1 × 0.9 = 0.99.

Key words

original amount: the amount before the change. It is 100%.

Back to your guess

The answer is C: Divide 150 by 1.25 to get 120.

The 25% rise was worked out on the old herd, not on 150. The new herd is 125% of the old herd, so divide 150 by 1.25. That gives 120. Check: 120 × 1.25 = 150.

Keep it

Copy this into your exercise book.

To go back from a new amount, divide by the multiplier. Never take the same percent off the new amount. A rise then a fall of 10% gives 0.99.

Draw: Two boxes joined by an arrow labelled × 1.2 above and a return arrow labelled ÷ 1.2 below.

M4.2.3 Chapter 3

Simple and compound interest

Two savings clubs both add 5% a year to the coins I put in. Why does one of them give more?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A One adds 5% of the start amount each year. The other adds 5% of the amount that has grown.
You picked A. Read on. At the end, the chapter comes back to your guess.
B They cannot differ. 5% a year is 5% a year.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The one that ends with more must have a higher rate.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Work out simple interest on 2 000 coins at 4% a year
Find the value after 3 years of 5% compound interest
Compare simple and compound interest in a table over 10 years

A saver puts 2 000 coins into a savings club. Each year the club adds interest, a percent of the savings. Here the rate is 4% a year.

Simple interest is the same amount every year. It is always 4% of the start amount. 4% of 2 000 is 80, so the saver gains 80 coins each year.

Simple interest on 2 000 coins at 4% a year. The interest is the same every year.

Worked example

Work out the simple interest on 2 000 coins at 4% a year for 5 years.

One year of interest: 4% of 2 000 = 0.04 × 2 000 = 80.
5 years of interest: 5 × 80 = 400.
The total is 2 000 + 400 = 2 400 coins.

Answer: 400 coins of interest

To find simple interest, multiply the start amount by the rate as a decimal, and then by the number of years.

Compound interest is different. Each year the interest is worked out on the whole amount, including interest already added. The amount is multiplied by the same multiplier every year. For 5% the multiplier is 1.05.

8 000 coins at 5% compound interest. The interest grows because the start of each year is bigger.

Worked example

8 000 coins earn 5% compound interest a year. How many coins are there after 3 years?

Each year the amount is multiplied by 1 + 0.05 = 1.05.
After year 1: 8 000 × 1.05 = 8 400.
After year 2: 8 400 × 1.05 = 8 820.
After year 3: 8 820 × 1.05 = 9 261.

Answer: 9 261 coins

The three steps multiply by 1.05 three times. That is 1.05 × 1.05 × 1.05, written 1.05³. So the value after n years is the start amount times the multiplier, n times over.

See it three ways

With things

Put in 1 000 coins at 10% a year. After one year add a tenth: 1 100. After year two add a tenth of 1 100: 1 210.

As a picture

In symbols

1 000 × 1.1 × 1.1 = 1 000 × 1.21 = 1 210.

Now compare the two over 10 years. Take 8 000 coins at 5% a year, both ways. In year 1 they are equal. After that, compound interest pulls ahead.

Draw it first

Simple and compound on one graph

1 Simple interest adds the same amount each year, so its graph is a straight line.
2 Compound interest adds more each year, so its graph bends upward. The curve starts on the line, then rises above it.

Compound values are rounded to the nearest coin. The two match in year 1, and then the gap widens each year.

Worked example

8 000 coins earn 5% a year for 10 years. How many more coins does compound interest give than simple interest?

Simple interest for 10 years: 8 000 × 0.05 × 10 = 4 000.
So the simple total is 8 000 + 4 000 = 12 000.
A calculator multiplies 8 000 by 1.05 ten times and gives 13 031, to the nearest coin.
Compound gives more by 13 031 − 12 000 = 1 031.

Answer: 1 031 coins more

Try it

Make your own table

You need: paper, a pencil.

Start with 1 000 coins and a rate of 10% a year. Draw three columns: year, simple and compound.
For simple interest, add 100 each year.
For compound interest, multiply the last amount by 1.1 each year. Round to the nearest coin.

What you will see: The two columns match in year 1. In year 2 compound is already ahead, with 1 210 against 1 200, and the gap grows.

If you cannot try it, this is what happens.

A wrong idea many people have

8 000 coins at 5% compound interest for 3 years earns 3 × 5% = 15%, so the total is 9 200.

Why it feels right: Three years of 5% looks like 15%, and it is the answer simple interest would give.

The test that breaks it: Work year by year: 8 000 → 8 400 → 8 820 → 9 261. The total is 9 261, which is more than 9 200.

What is true: Each year's interest is worked out on a bigger amount. Multiply by 1.05 three times, not by 1.15 once.

Key words

interest: extra amount added to savings each year, worked out as a percent.
simple interest: interest that is always a percent of the start amount, so it is the same every year.
compound interest: interest worked out on the amount so far, including interest already added.

Back to your guess

The answer is A: One adds 5% of the start amount each year. The other adds 5% of the amount that has grown.

The rate can be the same and the totals still differ. Simple interest is always worked out on the start amount. Compound interest is worked out on the amount that has grown, so it earns interest on interest.

Keep it

Copy this into your exercise book.

Simple interest: start × rate × years, the same each year. Compound interest: multiply by 1 + rate once for every year. Compound is ahead after year 1.

Draw: A graph with a straight line named simple and a curve named compound above it.

M4.2.4 Chapter 4

Growth and decay

The fish in a lake fall by 15% every year. Will the lake ever have no fish at all?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Yes, after about 7 years, because 15% a year for 7 years is about 100%.
You picked A. Read on. At the end, the chapter comes back to your guess.
B No. Each year it keeps 85% of the fish, so some always remain.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Yes, after exactly 15 years.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Use the multiplier 0.85 for a 15% fall each year
Model a goat herd that grows by 10% a year

Ana studies a lake. It has 8 000 fish. Each year the number falls by 15%. How many fish are left after 3 years?

A fall of 15% leaves 85% of the fish. As a decimal that is 0.85. Each year the lake keeps 0.85 of what it had. So each year you multiply by 0.85 again.

Draw it first

Watch the fish fall

1 At the start the lake has 8 000 fish, which is 8 on this scale of thousands.
2 After 1, 2 and 3 years the dots are lower each time. The drops get smaller, because 15% of a smaller number is smaller.
3 The curve joins the dots. It falls fast at first, then more slowly, and it stays above zero.

A quantity that falls by the same percent every year is in decay. A quantity that rises by the same percent every year is in growth. Both are described by a model: a rule that says how the amount changes.

Worked example

A lake has 8 000 fish. The number falls by 15% each year. How many fish are there after 3 years?

A fall of 15% leaves 85%, so the yearly multiplier is 0.85.
After year 1: 8 000 × 0.85 = 6 800.
After year 2: 6 800 × 0.85 = 5 780.
After year 3: 5 780 × 0.85 = 4 913.

Answer: 4 913 fish

Each row is 8 000 × 0.85 once for every year, rounded to the nearest whole fish. The fall is 1 200 in year 1 but only 533 in year 6.

Growth works the same way, with a multiplier above 1. A herd of 100 goats grows by 10% each year. The yearly multiplier is 1 + 0.1 = 1.1.

Growth bends upward. Each year the herd gains more than the year before.

Worked example

A herd of 100 goats grows by 10% each year. How many goats are there after 5 years, to the nearest goat?

A rise of 10% gives a yearly multiplier of 1.1.
Five years means multiplying by 1.1 five times: 1.1⁵ = 1.61051.
The herd is 100 × 1.61051 = 161.051.
Goats are whole animals, so round to the nearest goat: 161.

Answer: 161 goats

The model gives parts of a goat. We compare the model's numbers with 150 and only round at the end.

Worked example

The same herd starts at 100 goats. In which year does it first pass 150 goats?

Year 3 is below 150: 100 × 1.1³ = 133.1.
Year 4 is below 150: 133.1 × 1.1 = 146.41.
Year 5 is above 150: 146.41 × 1.1 = 161.051.
So the herd first passes 150 goats in year 5. In year 5 the model gives 161.051, which is about 161 goats.

Answer: Year 5

See it three ways

With things

A herd of 100 goats. After one year add a tenth: 110. After year two add a tenth of that: 121.

As a picture

In symbols

100 × 1.1 × 1.1 = 121.

Try it

Tear a strip of paper in half

You need: a strip of paper.

Let the whole strip stand for 100%. Tear it in half and keep one piece.
Tear the piece in half again and keep one piece. Do this three times in all.

What you will see: The pieces are 50%, 25% and 12.5% of the strip. Each tear multiplies by 0.5, a fall of 50%. In the model you can tear for ever and never reach nothing. A real strip becomes too small to tear.

If you cannot try it, this is what happens.

A wrong idea many people have

A fall of 15% a year takes about 7 years to wipe out the fish, because 7 × 15% is about 100%.

Why it feels right: Seven falls of 15 take off about 100. That is right for 15 fish a year, but not for 15 percent.

The test that breaks it: A fall of 15% takes 15% of what is left, which gets smaller every year. After 7 years the lake still has about 2 565 of its 8 000 fish, which is 32%.

What is true: Multiply by 0.85 once for every year: 0.85 × 0.85 and so on. The number keeps falling but never reaches zero.

A wrong idea many people have

A rise of 10% a year for 2 years is a rise of 20%.

Why it feels right: Two lots of 10% add up to 20%, and the sum looks like the rule for simple interest.

The test that breaks it: Work it out for 100 goats. After 2 years the herd is 121, not 120. The second rise is 10% of the bigger herd.

What is true: Multiply the multipliers: 1.1 × 1.1 = 1.21. That is a rise of 21%.

Key words

decay: a fall by the same percent every period. The multiplier is below 1.
growth: a rise by the same percent every period. The multiplier is above 1.
model: a rule that says how an amount changes, such as multiplying by 0.85 every year.

Back to your guess

The answer is B: No. Each year it keeps 85% of the fish, so some always remain.

Each year the lake keeps 85% of the fish it had. Multiplying a number above zero by 0.85 never gives zero. In the model the fish get fewer and fewer, but never none. A real lake has whole fish, so the model is a guide while there are many.

Keep it

Copy this into your exercise book.

A fall of p% a year multiplies by 1 − p/100 every year. A rise multiplies by 1 + p/100. Decay stays above zero. Growth bends upward.

Draw: A falling curve that flattens and a rising curve that steepens, each with its multiplier written beside it.

M4.2.5 Chapter 5

Harder ratio problems

Mei and Ana share beads in the ratio 2 : 3. Ana and Tariq share beads in the ratio 4 : 5. What is the ratio of all three?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Mei : Ana : Tariq is 2 : 3 : 5.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Mei : Ana : Tariq is 2 : 4 : 5.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Mei : Ana : Tariq is 8 : 12 : 15.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Combine two ratios into one, like a : b : c
Solve a ratio problem that changes when some are added

Mei and Ana share beads in the ratio 2 : 3. Ana and Tariq share beads in the ratio 4 : 5. Ana appears in both ratios. We want one ratio for all three.

Ana is 3 parts in the first ratio and 4 parts in the second. Those are different sizes of part, so we cannot join the numbers yet. First we make Ana the same in both.

Draw it first

Make the middle the same

1 Mei and Ana: 2 parts to 3 parts. Ana has 3 parts here.
2 Ana and Tariq: 4 parts to 5 parts. Ana has 4 parts here, a different size of part.
3 Cut each part into smaller equal pieces. Ana's bar keeps its length. Make Ana 12 pieces in both. Mei is then 8 pieces and Tariq is 15 pieces.

Multiplying both numbers of a ratio by the same number gives an equal ratio. 2 : 3 = 8 : 12 when you multiply by 4. Pick the numbers so Ana matches: the lowest common multiple of 3 and 4, which is 12.

Worked example

Mei : Ana is 2 : 3 and Ana : Tariq is 4 : 5. Find Mei : Ana : Tariq.

Ana has 3 parts in the first ratio and 4 parts in the second.
The lowest common multiple of 3 and 4 is 12. Make Ana 12 in both.
First ratio, multiply by 4: 2 : 3 = 8 : 12.
Second ratio, multiply by 3: 4 : 5 = 12 : 15.
Ana is 12 in both, so Mei : Ana : Tariq = 8 : 12 : 15.

Answer: 8 : 12 : 15

With a : b : c you can share an amount. Add the parts to find how many equal shares there are. Then find one part, and multiply.

70 beads in 35 parts. Each part is 2 beads.

Worked example

Mei, Ana and Tariq have 70 beads together in the ratio 8 : 12 : 15. How many beads does each have?

Add the parts: 8 + 12 + 15 = 35.
One part is 70 ÷ 35 = 2 beads.
Mei has 8 × 2 = 16, Ana has 12 × 2 = 24, and Tariq has 15 × 2 = 30.
Check: 16 + 24 + 30 = 70.

Answer: 16, 24 and 30 beads

Now a different kind of problem. Some are added, and the ratio changes. One group stays the same size. That group gives you the key.

The goats stay 3 parts. The sheep go from 2 parts to 5 parts. The dotted parts are the 12 sheep that were added.

Worked example

Kofi has goats and sheep in the ratio 3 : 2. He gets 12 more sheep, and now goats : sheep is 3 : 5. How many of each did he have at the start?

The goats did not change, so 3 parts of goats keep the same size.
The sheep went from 2 parts to 5 parts. That is 5 − 2 = 3 more parts.
The 3 more parts are 12 sheep, so one part is 12 ÷ 3 = 4.
At the start: goats 3 × 4 = 12, and sheep 2 × 4 = 8.
Check: after, 12 goats and 8 + 12 = 20 sheep, and 12 : 20 = 3 : 5.

Answer: 12 goats and 8 sheep

See it three ways

With things

Mei has 2 beads for every 3 of Ana's. Put 4 such sets together. Mei has 8 beads and Ana has 12.

As a picture

In symbols

2 : 3 = 8 : 12.

Try it

Check with stones

You need: 35 stones or beans.

Lay out 8 stones for Mei, 12 for Ana and 15 for Tariq, in three rows.
Group Mei's and Ana's stones in fours. Count the groups. Then group Ana's and Tariq's stones in threes, and count again.

What you will see: In fours, Mei has 2 groups and Ana has 3, so Mei : Ana is 2 : 3. In threes, Ana has 4 groups and Tariq has 5, so Ana : Tariq is 4 : 5.

If you cannot try it, this is what happens.

A wrong idea many people have

Mei : Ana is 2 : 3 and Ana : Tariq is 4 : 5, so Mei : Ana : Tariq is 2 : 3 : 5.

Why it feels right: The names Mei, Ana and Tariq come in order, and so it looks as if the numbers can come in order too.

The test that breaks it: In 2 : 3 : 5, Ana : Tariq would be 3 : 5. The question says Ana : Tariq is 4 : 5. That does not match.

What is true: The shared person must have the same number in both ratios. Make Ana 12 in both first, then join.

A wrong idea many people have

Goats : sheep is 3 : 2. Adding 12 sheep makes it 3 : 14.

Why it feels right: The sheep number in the ratio is 2, and 12 sheep are added, so 2 + 12 looks right.

The test that breaks it: The 2 in the ratio is 2 parts, not 2 sheep. A part is not the same size as one sheep. You cannot add 12 sheep to 2 parts.

What is true: Find how big one part is first. Here one part is 4 sheep, so 12 sheep are 3 parts.

Key words

lowest common multiple: the smallest whole number above 0 that both numbers divide into. For 3 and 4 it is 12.

Back to your guess

The answer is C: Mei : Ana : Tariq is 8 : 12 : 15.

Ana has 3 parts in one ratio and 4 parts in the other, so the numbers cannot just be joined. Make Ana the same in both: 12. Then 2 : 3 becomes 8 : 12, and 4 : 5 becomes 12 : 15. So Mei : Ana : Tariq is 8 : 12 : 15.

Keep it

Copy this into your exercise book.

To join a : b and b : c, make b the same in both with the lowest common multiple. When some are added, find what stayed the same, then the size of one part.

Draw: Two pairs of bars with the shared bar cut into 12 equal pieces, and the joined bars 8 : 12 : 15 beneath.

M4.2.6 Chapter 6

Proportion as a formula

If y is proportional to x, how do I write that as a formula?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A y = kx, where k is a fixed number.
You picked A. Read on. At the end, the chapter comes back to your guess.
B y = k + x, where k is a fixed number.
You picked B. Read on. At the end, the chapter comes back to your guess.
C y = k ÷ x, where k is a fixed number.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Write y = kx for direct proportion and find k
Solve problems where y is proportional to x² or to 1/x

A tap fills 5 litres every minute. After 1 minute there are 5 litres. After 2 minutes there are 10. After 3 minutes there are 15. Each time, the litres are the minutes times 5.

Draw it first

Every y is x times the same number

1 After 1 minute, y is 1 × 5 = 5.
2 After 2 minutes, y is 2 × 5 = 10.
3 After 3 minutes, y is 3 × 5 = 15.
4 The same number 5 multiplies every x. We write the rule as a formula: y = 5x.

We say y is directly proportional to x, or just proportional. It means y = kx, where k is a fixed number that never changes. The number k is the constant of proportionality. Here k = 5. It must not be 0.

To find k, take any pair of values and divide: k = y ÷ x. Then k stays the same for every pair.

The last column is always 5. That is how you know y is proportional to x.

Worked example

y is proportional to x. When x = 7, y = 35. Find the formula, and find y when x = 12.

Write the formula y = kx.
Put in x = 7 and y = 35: 35 = k × 7, so k = 35 ÷ 7 = 5.
The formula is y = 5x.
When x = 12, y = 5 × 12 = 60.

Answer: y = 5x. When x = 12, y = 60.

Some things grow faster than that. Take a right-angled triangle whose two shorter sides are both x cm. Its area is half of x times x. When x doubles, the area does not double. It becomes four times as big.

Double the side and the area is 2 × 2 = 4 times as big.

When y is proportional to x², the formula is y = kx². For the triangle, y = 0.5x². Doubling x multiplies y by 2² = 4.

The area y cm² of a right-angled triangle with two sides of x cm. Going from x = 2 to x = 4 multiplies y by 4. Going to x = 6 multiplies it by 9.

A right-angled triangle with two sides of 6 cm is half of a 6 cm by 6 cm square. The square holds 36 tiles of 1 cm², so the triangle has an area of 18 cm².

Worked example

A right-angled triangle has two equal shorter sides of x cm. Its area y cm² is proportional to x². When x = 6, y = 18. Find y when x = 10.

Write the formula y = kx².
Put in x = 6 and y = 18: 18 = k × 6², so k = 18 ÷ 36 = 0.5.
The formula is y = 0.5x².
When x = 10, y = 0.5 × 10² = 0.5 × 100 = 50.

Answer: y = 0.5x². When x = 10, y = 50.

The other kind goes the opposite way. A 120 km journey takes 3 hours at 40 km per hour. At twice the speed it takes half the time. The time y hours is 120 divided by the speed x.

We say y is inversely proportional to x, or proportional to 1/x. The formula is y = k ÷ x. Multiply x and y and you always get k. Here x is a speed, so it is above 0.

Speed is in km per hour. The last column is always 120 km. When the speed goes up, the time goes down.

Worked example

The time y hours for a journey is inversely proportional to the speed x km per hour. At 40 km per hour it takes 3 hours. How long does it take at 60 km per hour?

Time is proportional to 1 ÷ speed, so y = k ÷ x.
At 40 km per hour the time is 3 hours: 3 = k ÷ 40.
Multiply both sides by 40: k = 3 × 40 = 120.
The formula is y = 120 ÷ x. Check: 120 ÷ 40 = 3.
At 60 km per hour: y = 120 ÷ 60 = 2.

Answer: 2 hours

See it three ways

With things

A tap fills 5 litres each minute. After 1 minute: 5 litres. After 2 minutes: 10. After 3 minutes: 15.

As a picture

In symbols

y = 5x.

Try it

Rectangles of the same area

You need: squared paper, a pencil.

Draw a rectangle 1 square across and 12 squares up. It holds 12 squares.
Draw more rectangles that also hold 12 squares: 2 across, 3 across, 4 across. Write how many squares go up each time.

What you will see: As the number across goes up, the number up goes down. Across times up is always 12. That is inverse proportion.

If you cannot try it, this is what happens.

A wrong idea many people have

If y is proportional to x², then doubling x doubles y.

Why it feels right: Proportional usually means "when one doubles, the other doubles", and that is how direct proportion works.

The test that breaks it: Take y = 0.5x². When x = 2, y = 2. When x = 4, which is double, y = 8. That is four times as big, not double.

What is true: When y is proportional to x², doubling x multiplies y by 2² = 4. Tripling x multiplies y by 9.

A wrong idea many people have

Inverse proportion means y goes down by the same amount that x goes up.

Why it feels right: One goes up and the other goes down, so it feels as if they trade places evenly.

The test that breaks it: Take y = 24 ÷ x. When x = 4, y = 6. When x = 8, y = 3, which is a fall of 3. When x = 12, y = 2, which is a fall of only 1.

What is true: When x is multiplied by a number, y is divided by the same number. Then x times y stays the same.

Key words

directly proportional: y = kx for a fixed number k. Doubling x doubles y.
constant of proportionality: the fixed number k in y = kx or y = k ÷ x.
inversely proportional: y = k ÷ x for a fixed number k. Doubling x halves y.

Back to your guess

The answer is A: y = kx, where k is a fixed number.

Proportional means y is always the same multiple of x. Double x and y doubles. So y = kx, with k a fixed number. The formula y = k ÷ x is for inverse proportion, where doubling x halves y.

Keep it

Copy this into your exercise book.

Proportional to x: y = kx, and y ÷ x is always k. Proportional to x²: y = kx². Inversely proportional to x: y = k ÷ x, and x times y is always k.

Draw: Three small tables, one for each kind, with the column y ÷ x, y ÷ x² or x × y beside each.

M4.2.7 Chapter 7

Graphs of proportion

How can I tell from a graph that two amounts are in proportion?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A The graph is any straight line.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The graph is a straight line that goes through the origin.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The graph is a curve that goes down.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Sketch the graphs of y = kx and y = k/x
Spot direct or inverse proportion from a table or graph

Ravi has a tap that fills 5 litres every minute. He plots minutes across and litres up. The rule is y = 5x.

Draw it first

Plot the tap

1 Plot a dot for each minute: (1, 5), (2, 10), (3, 15), (4, 20), (5, 25).
2 The dots lie in a straight line. Draw the line through them.
3 At 0 minutes there is 0 litres, so the line starts at (0, 0), the origin.

The graph of y = kx is always a straight line through the origin, the point (0, 0). The number k says how steep the line is. It is the gradient: how much y goes up for each 1 across. Here k = 5, so y goes up 5 for each 1 across.

A bigger k gives a steeper line. All three lines pass through the origin.

Worked example

Sketch the graph of y = 3x for x from 0 to 4.

Choose some values of x: 0, 1, 2, 3 and 4.
Work out y = 3x for each: y = 0, 3, 6, 9 and 12. The last is y = 12.
Plot the points (0, 0), (1, 3), (2, 6), (3, 9) and (4, 12).
Join them with a straight line. It goes through the origin.

Answer: A straight line through (0, 0) and (4, 12)

The graph of y = 3x.

A straight line is not enough. Compare y = 2x with y = 2x + 3. The second line is straight, but it crosses the y axis at 3, not at 0. When x = 0 it gives y = 3. Doubling x does not double y, so it is not proportion.

Both lines are straight. Only the one through the origin is proportion.

Now inverse proportion. Take y = 12 ÷ x, for x above 0. When x is 1, 2, 3, 4, y is 12, 6, 4, 3. The points do not lie in a line. They lie on a curve that falls.

See it three ways

With things

Rectangles that hold 12 squares: 1 across and 12 up, 2 across and 6 up, 3 across and 4 up.

As a picture

In symbols

x × y = 12, so y = 12 ÷ x.

The curve gets very high near the y axis, and very close to the x axis far along. It comes as near as you like to each axis but never touches either. A y of 0 would need 12 ÷ x = 0, and no x gives that.

Worked example

Sketch the graph of y = 12 ÷ x for x from 1 to 12.

Choose values of x that divide 12: 1, 2, 3, 4, 6 and 12.
When x = 2, y = 12 ÷ 2 = 6.
When x = 4, y = 12 ÷ 4 = 3.
When x = 6, y = 12 ÷ 6 = 2.
Plot the points and join them with a smooth curve. It falls and never touches the axes.

Answer: A smooth curve through (2, 6), (4, 3) and (6, 2)

You can also tell the two kinds apart from a table. For direct proportion, y ÷ x is the same every time. For inverse proportion, x × y is the same every time.

Test a table: divide, then multiply. One column stays the same.

Worked example

A table has x = 2, 4, 6, 8 and y = 3, 6, 9, 12. Is y proportional to x, inversely proportional, or neither?

Divide y by x for the first pair: 3 ÷ 2 = 1.5.
Divide for the others: 6 ÷ 4, 9 ÷ 6 and 12 ÷ 8 are all 1.5.
Now multiply: 2 × 3 = 6, but 4 × 6 = 24. This is not fixed.
y ÷ x is always 1.5, so y is proportional to x, with k = 1.5.

Answer: Directly proportional, with k = 1.5

Try it

Plot y = 2x on squared paper

You need: squared paper, a ruler, a pencil.

Draw an x axis from 0 to 4 and a y axis from 0 to 8.
Plot the points (0, 0), (1, 2), (2, 4), (3, 6) and (4, 8).
Lay the ruler along the points and draw the line.

What you will see: The points lie on one straight line, and the line passes through the origin. Each step across goes up 2.

If you cannot try it, this is what happens.

A wrong idea many people have

Any straight line shows direct proportion.

Why it feels right: Direct proportion makes a straight line, so a straight line feels like proof.

The test that breaks it: Take y = 2x + 3. When x = 1, y = 5. When x = 2, which is double, y = 7. That is not double 5.

What is true: Direct proportion needs a straight line through the origin. Zero of x must give zero of y.

A wrong idea many people have

If y goes down as x goes up, the graph shows inverse proportion.

Why it feels right: Inverse proportion does go down as x goes up, so any fall looks like it.

The test that breaks it: Take y = 10 − x. It falls as x goes up. When x = 2, y = 8, and x × y = 16. When x = 4, y = 6, and x × y = 24. The product is not fixed.

What is true: Inverse proportion needs x × y to stay the same. Its graph is a curve that never touches the axes, not a straight line.

Key words

origin: the point (0, 0), where the two axes cross.
gradient: how steep a line is: how much y goes up for each 1 across.

Back to your guess

The answer is B: The graph is a straight line that goes through the origin.

Direct proportion makes a straight line through the origin, the point where both amounts are zero. A straight line that misses the origin is not proportion. A curve that falls and never touches the axes shows inverse proportion.

Keep it

Copy this into your exercise book.

The graph of y = kx is a straight line through the origin. The graph of y = k ÷ x is a curve that never touches the axes. A table: y ÷ x fixed is direct, x × y fixed is inverse.

Draw: A straight line through the origin beside a falling curve, with y ÷ x and x × y written under them.

Add the rise to the start

Cut a strip

Work back from the new amount

Try any number

Simple and compound on one graph

Make your own table

Watch the fish fall

Tear a strip of paper in half

Make the middle the same

Check with stones

Every y is x times the same number

Rectangles of the same area

Plot the tap

Plot y = 2x on squared paper

Key words, A to Z