Fractions, Percentages and Proportions

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Fractions, Percentages and Proportions

In this lesson, you can form proportional noun phrases and identify their whole.

Created by , British English teacher.

I’m studying:

Try the activities, then check your answers.

Choose a short recording.

Before you start

Say how many three quarters of eight are.

See one possible response

Six.

You explain the results of a small fictional club survey and the supplies left after an event.

What you will practise
  • Form proportional noun phrases and identify their whole.
  • Choose agreement for straightforward plural groups and uncountable amounts.
  • Explain a proportion without inventing a total or changing the comparison group.

Try each response before revealing a model. Equivalent grammatical answers are welcome when they preserve the task meaning.

Part of a named whole

Use a fraction or percentage with of to identify the whole: half of the guests; sixty per cent of the paper. The whole must be clear. A percentage alone does not tell you the number of people unless the total is supplied.

Half of the twelve guests arrived early.

Twenty-five per cent of the forty tickets are still available.

They discuss volunteer availability using exactly ten identical circular counters on a board: five blue and five pale grey, arranged in two rows of five. A blank calendar page supports Friday planning. Only two actual adults; the ten counters represent volunteers.
Half of our ten volunteers are free on Friday.Five coloured circles out of ten represent the available half.

Agreement follows what is measured

With a plural group, use plural agreement: half of the chairs are outside. With an uncountable amount, use singular agreement: half of the water is left. Do not decide from the fraction word alone.

A third of the books are new.

Half of the paint is unused.

Keep the size and interpretation honest

Half of ten people means five people; half of the group without a total does not give an exact number. Use the same identified whole when comparing proportions. One half is not necessarily a large absolute number.

Half of our ten volunteers are free on Friday.

Most of the room’s space is clear, but only a quarter of the shelves are empty.

A supplied total turns a percentage into a count

25%one quarter40 tickets ÷ 410 tickets
Twenty-five per cent of the forty tickets are still available. The total forty lets us calculate ten. Without the total, the percentage alone would not give the number of tickets.
They consider a paint pot half filled with unused paint, with one clean paint roller, closed second lid and a graduated clear measuring cup without written marks. The amount is half of one uncountable paint supply, not half the adults.
Half of the paint is unused.Use singular is with this uncountable amount.

Guided practice

Say or write your answer first. Then compare the meaning and grammar with the model.

1. Half of the chairs ___ ready.

Compare a model and explanation

are

Chairs is a plural group.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

2. Half of the rice ___ left.

Compare a model and explanation

is

Rice is an uncountable amount.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

3. A quarter of twenty tickets remain. How many?

Compare a model and explanation

Five tickets.

Twenty divided into four equal parts gives five.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

4. Two thirds of the books is new.

Compare a model and explanation

Two thirds of the books are new.

The measured items are plural books.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

5. Thirty per cent of the workshop’s seats are empty. What is the whole?

Compare a model and explanation

All the seats in that workshop.

Of identifies the comparison set.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

6. Half the members attended. The total is unknown. Can you conclude exactly ten attended?

Compare a model and explanation

No.

The proportion does not supply the membership total.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

7. Half of the water is left. Half of the bottles are full. Why do the verbs differ?

Compare a model and explanation

Water is an amount; bottles are plural items.

Agreement follows the relevant complement.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

8. Eight people answered a survey; four chose Tuesday and two chose Thursday. Give the two proportions.

Compare a model and explanation

Half of the respondents chose Tuesday, and a quarter chose Thursday.

Both proportions use the same total of eight.

Accept a grammatical alternative that satisfies the stated prompt and this item’s explanation; the model is not an exhaustive exact-string key.

A short conversation

Listen to the exchange, then take one role. The speaker labels identify roles; do not narrate them.

Open the conversation transcript

Organiser: How many people answered our survey?

Helper: Twenty. Half of them chose Saturday.

Organiser: So ten people. Is there enough paper for the activity?

Helper: Yes. Half of the paper is still unused.

Listening checks

Listen to each extract, then choose the best answer. These three questions have their own score.

1. How many chose the morning?
Read the listening transcript

Half of the sixteen participants chose the morning session.

2. What is described as missing?
Read the listening transcript

Half of the paint is left, but half of the brushes are missing.

3. What can be stated exactly from this message?
Read the listening transcript

Thirty per cent of the members replied. We don’t know how many members there are.

Answer key with explanations

  1. How many chose the morning?

    Eight.: Half of sixteen is eight.

    Compare all choices
    • Eight.: Half of sixteen is eight.
    • Sixteen.: That is the whole group.
    • Four.: Four is a quarter, not half.

  2. What is described as missing?

    Brushes.: Missing applies to the plural brushes.

    Compare all choices
    • The participants.: They are not discussed.
    • Brushes.: Missing applies to the plural brushes.
    • All the paint.: Half the paint is left, not all missing.

  3. What can be stated exactly from this message?

    The reply proportion is thirty per cent.: The percentage is given, while the total is unknown.

    Compare all choices
    • Exactly thirty members replied.: That would require a total of one hundred, which is not supplied.
    • Nobody replied.: The stated proportion is positive.
    • The reply proportion is thirty per cent.: The percentage is given, while the total is unknown.

Mastery quiz

Choose the best answer for each situation. The score is out of eight.

1. Complete: Half of the forms ___ complete.
2. Complete: A third of the milk ___ left.
3. A quarter of twelve seats are empty. How many?
4. What does of the guests identify?
5. Half the members agree. No total is given. What is not known?
6. Choose the agreement pair.
7. Twenty-five per cent of forty is how many?
8. Which comparison uses a clear common whole?
Answer key with explanations

  1. Complete: Half of the forms ___ complete.

    are: Forms is a plural group.

    Compare all choices
    • is: The target measured group is plural.
    • be: A finite present verb is needed.
    • are: Forms is a plural group.

  2. Complete: A third of the milk ___ left.

    is: Milk is an uncountable amount.

    Compare all choices
    • is: Milk is an uncountable amount.
    • have: This does not form the intended predicate.
    • are: The amount is not a plural count noun here.

  3. A quarter of twelve seats are empty. How many?

    Three.: Twelve divided by four is three.

    Compare all choices
    • Four.: Four is a third of twelve.
    • Three.: Twelve divided by four is three.
    • Twelve.: That is the full set.

  4. What does of the guests identify?

    The whole group the proportion concerns.: The of phrase establishes the comparison set.

    Compare all choices
    • The exact number of guests without any total.: An identified whole is not the same as a supplied numerical total.
    • A different unmentioned group.: It names the guests, not an unknown substitute.
    • The whole group the proportion concerns.: The of phrase establishes the comparison set.

  5. Half the members agree. No total is given. What is not known?

    The exact number agreeing.: The proportion alone does not give a count.

    Compare all choices
    • The exact number agreeing.: The proportion alone does not give a count.
    • That the stated proportion is half.: That is explicit.
    • Whether the sentence refers to members.: Members is explicitly named.

  6. Choose the agreement pair.

    Half of the water is left; half of the cups are clean.: The amount is singular and the counted group plural.

    Compare all choices
    • Half of the water is left; half of the cups is clean.: The second clause needs are because cups is plural in this proportion.
    • Half of the water is left; half of the cups are clean.: The amount is singular and the counted group plural.
    • Half of the water are left; half of the cups is clean.: Both agreement choices are reversed for these straightforward nouns.

  7. Twenty-five per cent of forty is how many?

    Ten.: A quarter of forty is ten.

    Compare all choices
    • Twenty-five.: The percentage is not automatically the absolute count.
    • Forty.: That is the total, not the quarter.
    • Ten.: A quarter of forty is ten.

  8. Which comparison uses a clear common whole?

    Half of the same ten volunteers chose Friday; a fifth chose Monday.: Both proportions refer to the named ten.

    Compare all choices
    • Half of the same ten volunteers chose Friday; a fifth chose Monday.: Both proportions refer to the named ten.
    • A quarter is always a large crowd.: Absolute size depends on the total.
    • Half chose Friday, so exactly fifty people chose it.: No hundred-person total is supplied.

Use it in your own words

A small survey

Invent a total of eight or ten respondents and describe one simple fraction of it.

Reveal a possible response

Eight people answered. Half of them preferred the morning, so that’s four people.

  • Give the total
  • Keep the proportion and number consistent

Keep arithmetic simple.

Supplies remaining

Describe an amount of material and a group of objects with different agreement.

Reveal a possible response

Half of the paper is unused. A third of the pens are missing.

  • Use singular for the amount
  • Use plural for counted items

Other familiar supplies are welcome.

Put it all together

Give a short fictional event update: a known proportion of volunteers, a remaining amount of material and a limitation where the total is unknown.

Reveal a possible response

Half of our twelve volunteers can help, so six people are available. Half of the paint is left. Thirty per cent of the wider club replied, but I don’t know the membership total.

  • Combine numbers and agreement
  • Keep comparison groups distinct
  • State the unknown count honestly

Different simple figures are acceptable if internally consistent.

Take it with you

A proportion needs a clear whole. Agreement follows what the proportion refers to in these patterns. A proportion alone does not give an exact headcount without the total.