Math // Ratio & Proportion

Dividing Quantities by Ratio

Divide a given quantity into two parts in a given part:part or part:whole ratio, and express the division as a fraction of the whole

Ages
11 to 13
Type
Method
Builds on
4 topics
Leads to
1 topic
Standards
2 codes

Quick check

If £60 is shared between two people in the ratio 2:3, can the learner work out how much each person gets?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Share £60 between two people in the ratio 2:3 by finding the value of one part
  • Express the result of sharing 24 sweets as 10 and 14 in ratio form as 5:7
  • Solve problems involving three-part ratios (e.g. divide 180 in the ratio 1:2:3)

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

4 topics
  • Bar Models for RatiosRequired

    Ratio & Proportion // Ages 9 to 12

    Dividing a quantity in a given ratio is the natural next step from the bar model representation

  • Understanding fractionsRequired

    Ratio & Proportion // Ages 10 to 11

    Dividing quantities in a ratio extends KS2 unequal sharing problems

  • Ratio NotationRequired

    Ratio & Proportion // Ages 11 to 12

    Dividing in a ratio requires understanding ratio notation and simplification first

  • Proportional Reasoning VocabularyRequired

    Ratio & Proportion // Ages 11 to 14

    Dividing in a given ratio requires 'ratio', 'part-to-part', and 'part-to-whole' vocabulary

You are here

Dividing Quantities by Ratio

Ages 11 to 13 // Method

Leads to

1 topic
  • ProportionRequired

    Ratio & Proportion // Ages 12 to 14

    Proportion problems require dividing quantities in given ratios

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Divide a quantity in a given ratio using the find-one-part routine: total parts = sum of ALL ratio numbers; one part = quantity ÷ total parts; each share = its ratio number × one part. The guidance generalises this to three-part ratios (divide 180 in 1:2:3 = 6 parts, not 3) and works both directions: express a completed division back as a simplest-form ratio (10 and 14 → 5:7) and state a share as a fraction of the whole. The persistent hazard is using one part, rather than the sum of parts, as the denominator; a bar/tape model with the correct number of equal segments keeps the correct total visible as a check.

Common misconceptions

  • Dividing by a single part rather than the sum of parts (÷3 or ÷5 instead of ÷8), or in a 3-part ratio forgetting to include every part in the total.
  • Sharing the quantity equally among the number of terms (180 into three equal 60s for 1:2:3).
  • When reversing, writing the shares as a ratio without simplifying (10:14 left un-reduced), or reversing the order.
  • Confusing part-to-part with part-to-whole — stating a share as a/b of the whole rather than a/(a+b).

Useful representations

  • Bar model / tape diagram with a+b(+c) equal segments
  • The 'one part' unit (quantity ÷ total parts)
  • Ratio table (parts | value) extended to three columns for three-part ratios
  • Simplest-form ratio from actual shares (divide by the HCF)

Teaching practices

  • Standardise the routine explicitly: total parts = sum of all ratio numbers → one part = quantity ÷ total → multiply each ratio number by one part.
  • Extend to three-part ratios immediately (1:2:3 → 6 parts) so the 'sum ALL the parts' step is generalised, not tied to two shares.
  • Practise the reverse direction: given the actual shares, write them as a ratio and simplify (10:14 → 5:7), and state a share as a fraction of the whole.
  • Keep the bar model as the check: the number of equal segments equals the sum of the ratio numbers; adding the shares must return the original quantity.
  • Distinguish part-to-part (a:b) from part-to-whole (a out of a+b) in the notation, since the same numbers appear in both.

Activities to try

  1. 1

    Ratio-divider on a tape diagram (forward direction)

    Why it works: A three-part ratio 1:2:3 needs six equal parts, not three; dividing once by the total number of parts and then scaling each term is the whole routine, and the shares must add back to the original quantity.

    1. Enter a ratio such as 1:2:3 and cut the strip into the SUM of the parts — six equal segments for 1:2:3, not three.
    2. Let the quantity (e.g. 180) distribute across the equal segments and read off one part = 180 ÷ 6 = 30.
    3. Colour 1, 2 and 3 adjacent segments to read the shares 30, 60 and 90, then add them to check they return 180.
    4. Repeat with a two-part ratio (e.g. share £60 in 2:3: 5 parts, one part = 12, shares 24 and 36, summing to 60).
  2. 2

    Reverse station: write shares as a simplest-form ratio

    Why it works: A completed division can be expressed back as a ratio, and a ratio must be reduced to simplest form by dividing by the common factor — directly analogous to simplifying a fraction.

    1. Take two actual shares (e.g. 24 sweets shared as 10 and 14) and write them as a ratio 10:14.
    2. Simplify by dividing both terms by their highest common factor (÷2) to reach 5:7, keeping the order matching the shares as stated.
    3. Explain how this is like simplifying a fraction, and check you have not reversed the order (14:10 is wrong for 10 then 14).
  3. 3

    Part of the whole as a fraction

    Why it works: When stating a share as a fraction of the whole, the denominator is the SUM of all parts (a+b+c), not another single term; part-to-part and part-to-whole use the same numbers differently.

    1. For 1:2:3, state what fraction of the whole each share is: 1/6, 2/6 (=1/3) and 3/6 (=1/2), using the TOTAL of 6 parts as the denominator.
    2. Contrast part-to-part with part-to-whole explicitly: the middle share is 2 parts out of 6 total, so 2/6, NOT 2/3.
    3. Optionally toggle a colour band to light each part and reveal its fraction of the whole.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
6.RP.3
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Ratio.5

Math // Ratio & Proportion

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