Year 7 · Unit 2 Percentages · Lesson 2 of 6

One number, three names

3/8, 0.375 and 37.5% are the same amount. Slide a fraction onto a decimal line and a percentage line, then put mixed numbers in order.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 1 of this unit) A bar is cut into 20 equal parts and 6 are shaded. What percentage is shaded?
    Answer30%. Each part is 100 ÷ 20 = 5%, and 6 × 5 = 30.
  2. (Unit 1, lesson 1) Which is greater, 0.4 or 0.35?
    Answer0.4, because the tenths are 4 and 3. Write 0.4 as 0.40 and compare.
  3. (Unit 1, lesson 5) Write 12/18 in simplest form.
    Answer2/3. The greatest common factor of 12 and 18 is 6.
Words used in this lesson
fraction
a number written as one number over another. For example: 3/8.
decimal
a number written with a decimal point. For example: 0.375.
percentage
a number out of 100. For example: 37.5%.
equivalent
equal in value, written differently. For example: 1/2, 0.5 and 50% are equivalent.
simplest form
a fraction where the top and bottom have no common factor but 1. For example: 45/100 in simplest form is 9/20.
benchmark
a value worth knowing by heart, to build others from. For example: 1/8 = 12.5%.

One amount, three names

A shop in Accra shows one price three ways: the sign says “half price”, the till slip says “price × 0.5” and the website says “50% off”. Three notices, one price. A fraction, a decimal and a percentage are three names for the same amount, and each is useful for something different: fractions for sharing, decimals for measuring, percentages for comparing shares of different wholes.

Predict first: 3/8 of the bar is shaded. Is that more or less than 1/3 of the bar? Roughly what percentage? Then slide and read all three scales.

8
3
3/8fraction
0.375decimal
37.5%percentage

3/8 of the bar is shaded. On the decimal line it is 0.375 and on the percentage line it is 37.5%: one length, three names.

Show me the steps
  1. 1/2 is 0.5 on the decimal line and 50% on the percentage line.
  2. 3/4 is 0.75 and 75%.
  3. 1/5 is 0.2 and 20%.
  4. 3/8 is 0.375 and 37.5%: the eighth is 12.5%, so three of them are 37.5%.
  5. 7/40 is 0.175 and 17.5%.
  6. 6/20 is 0.3 and 30%, and in simplest form 6/20 is 3/10.

The same amount has a fraction, a decimal and a percentage name. A fraction is a ÷ b. A percentage is the decimal × 100. So 3/8 = 3 ÷ 8 = 0.375 = 37.5%.

It works past 1: 5/4 = 5 ÷ 4 = 1.25 = 125%, which is more than the whole.

Three ways in: scale the denominator to 100 when you can (7/20 = 35/100 = 35%); divide the top by the bottom; or use a benchmark you know (1/8 = 12.5%, so 3/8 = 37.5%).

Going back: a percentage is a number over 100 (45% = 45/100), so divide the top and bottom by their greatest common factor: 45/100 = 9/20.

Which is bigger?

To put numbers in different forms in order, do not compare the digits. Change all of them to one form first, most often percentages, or place them on a line. Move through the six sets.

1
0.35 < 2/5 < 41%smallest to largest
35% < 40% < 41%as percentages

2/5 is 40%, 0.35 is 35% and 41% is 41%, so in order: 0.35 < 2/5 < 41%.

Show me the steps
  1. Set 1: 2/5 is 40%, 0.35 is 35% and 41% is 41%. In order: 0.35 < 2/5 < 41%.
  2. Set 2: 3/4 is 75%, 0.7 is 70% and 72% is 72%. In order: 0.7 < 72% < 3/4.
  3. Set 3: 1/8 is 12.5%, 0.12 is 12% and 13% is 13%. In order: 0.12 < 1/8 < 13%.
  4. Set 4: 5/8 is 62.5%, 0.6 is 60% and 61% is 61%. In order: 0.6 < 61% < 5/8.
  5. Set 5: 7/20, 0.35 and 35% are all 35%, so all three are equal.
  6. Set 6: 3/5 is 60%, 0.06 is 6% and 6% is 6%. In order: 0.06 = 6% < 3/5.

A common mistake: using the digits of one form as the digits of another. 1/4 is 25%, not 14% or 4%. 0.07 is 7%, not 70%. Check with a benchmark: 1/4 is a quarter, and a quarter of 100 is 25.

Worked example: three forms

Write 7/20 as a decimal and as a percentage.

  1. 20 × 5 = 100, so multiply the top and bottom by 5: 7/20 = 35/100.
  2. 35 out of 100 is 35%.
  3. As a decimal, 35/100 = 0.35, so 7/20 = 0.35 = 35%.

Your go. The first step is given. Put 0.45, 3/5 and 43% in order, smallest first.

  1. Change each to a percentage: 0.45 = 45%, 3/5 = 60%, and 43% stays 43%.
  2. Now put 45%, 60% and 43% in order yourself, then check.
Check the order

43% < 45% < 60%, so in the original forms: 43% < 0.45 < 3/5.

Check before you go on

1. Which of these is the same as 0.07?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Write 3/5 as a percentage.

3. Which fraction in its simplest form is 40%?

Level 2 · Building

4. Write 9/25 as a percentage.

5. Write 62.5% as a decimal. Give your answer to three decimal places.

6. Which list is in order, smallest first?

Level 3 · Stretch

7. A water tank is 3/8 full and a second tank is 36% full. How many more per cent of a full tank does the first tank hold? Give your answer to one decimal place.

8. The fraction a/20 is equal to 85%. Find a.

Something to think about

1/3 as a decimal is 0.333... with the 3s going on for ever. What percentage is that, and why can a hundred grid never show exactly 1/3?

One way to think about it

1 ÷ 3 = 0.333..., so 1/3 is 33.3...%, which is 33 and a third per cent. A hundred grid has 100 squares and 100 is not a multiple of 3, so no number of whole squares is exactly a third: 33 squares is a little less and 34 a little more.

Exit ticket

9. Sana says 1/4 is 4%, because it has a 4 in it. Which reply is right?