lesson
Part to part is not part to whole
1 · Learn
A ratio compares quantities in a stated order. Red:blue = 12:18 simplifies to 2:3 by dividing BOTH parts by 6. It does not mean there are only five objects: it describes how the quantities are related. Equivalent ratios multiply or divide all parts by the same non-zero factor.
To share 60 counters in red:blue = 2:3, the whole has 2+3 = 5 equal ratio parts. One part is 60÷5 = 12 counters. Red gets 2×12 = 24 and blue gets 3×12 = 36. Check both the total, 24+36 = 60, and the simplified ratio, 24:36 = 2:3.
The fraction of ALL counters that is red is 2/5, not 2/3. The fraction comparing red with BLUE is 2/3. In the same arrangement, red:total is 2:5. Always name the second quantity before turning a ratio into a fraction.
Sometimes one part, rather than the total, is known. If red:blue is 2:3 and red is 18, then each ratio part is 18÷2 = 9. Blue is 3×9 = 27 and the total is 45. Dividing 18 by 5 would wrongly treat the red count as the whole.
A part-to-whole statement can be turned back into a part-to-part ratio. If 3/8 of a collection is red and all the rest is blue, 5/8 is blue, so red:blue = 3:5. This assumes the two groups include the whole collection without overlap. Match units first if a ratio compares measurements.
2 · Worked example
Share a 48 cm ribbon into lengths in ratio 3:5. There are eight parts, each 6 cm. The lengths are 18 cm and 30 cm. The first is 3/8 of the whole but 3/5 of the second length.
3 · Your turn
Simplify 15:25. Share 64 counters in that simplified ratio. What fraction of the whole is in the first group? If the first group instead had 21 counters, how many would the second group have?
Check your answer
3:5. Eight parts give 8 counters per part, so 24 and 40. The first group is 3/8 of the whole. With first group 21, one part is 7, so the second has 35.
4 · Apply your learning
Use five equal-sized boxes to represent 2:3. Label the boxes for a total of 60, then for a first-group count of 18. Explain why the two starting numbers require different first divisions.
Adult guidance and safety
Ask pupils to identify the reference whole and explain each equivalent representation. Keep answers exact until rounding is explicitly requested. Use counters or a bar diagram for sharing before a symbolic rule.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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