---
title: "Reasoning with Equivalences"
description: "Recognise and use repeated reasoning to generalise: extend patterns in equivalent fractions and percentage conversions, derive unknown facts from known facts, describe general rules for sequences and "
canonical: https://lightmysky.com/learn/mathematics/reasoning-with-equivalences-mt_jCy07DyBNU
source: https://lightmysky.com/learn/mathematics/reasoning-with-equivalences-mt_jCy07DyBNU.md
retrieved: 2026-10-19
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Reasoning with Equivalences

Recognise and use repeated reasoning to generalise: extend patterns in equivalent fractions and percentage conversions, derive unknown facts from known facts, describe general rules for sequences and predict terms

Subject: Mathematics · Area: Mathematical Thinking · Ages 9 to 10
Page: https://lightmysky.com/learn/mathematics/reasoning-with-equivalences-mt_jCy07DyBNU

## Ready when they can

- Notice that multiplying any number by 25 can be done by multiplying by 100 then dividing by 4, and explain why
- Describe the general rule for a sequence and predict the 20th term
- Generalise: to find 10% divide by 10, to find 5% halve 10%, and use this to find 35% of any number

## Lesson: Same Rule, Every Time

When two forms match exactly, state the rule in words and check why the match holds, then use it for fractions, sequences, or percentages. For example, 100 divided by 4 is 25, so x100 then ÷4 matches x25.

*(drawing: Each step adds 1 on top and 2 on the bottom. The value stays 1/2. What comes next?)*

**Example.** Mia's sequence starts at 5 and adds 3 each time: 5, 8, 11, 14, and so on. To find the 20th term, she does not need to write out 20 numbers. She uses the rule: start at 5, then add 3 a total of 19 times. That gives 5 + (3 × 19) = 62.

**Example.** To find 10% of 40, divide by 10: that's 4. Since 5% is half of 10%, 5% of 40 is 2. Adding 30% (three lots of 10%) and 5% gives 35%: 12 + 2 = 14. So 35% of 40 is 14.

Multiplying by 25 is the same as multiplying by 100, then dividing by 4. That works because 25 is 100 divided by 4, so the two steps together do exactly what ×25 does. Try it: 8 × 25 = (8 × 100) ÷ 4 = 800 ÷ 4 = 200.

*(drawing: 10% of 60 is 6. Half is 3, so 5% is 3. Three 10% chunks plus one 5% chunk make 35%.)*

100 divided by 4 is 25. So x100 then ÷4 matches x25. For 6 x 25: 6 x 100 = 600, then 600 ÷ 4 = 150. If the rule holds, the match is exact.

**Recap.** Find a rule that works once, explain why it works, then use it again for tricky multiplications, sequences, and percentages.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Generating a pattern from a rule](https://lightmysky.com/learn/mathematics/generating-a-pattern-from-a-rule-mt_A-FyLLLLzy)
- [Generalising from repeated reasoning](https://lightmysky.com/learn/mathematics/generalising-from-repeated-reasoning-mt_aivrWs6jrS)
- [Equivalent fractions (age 9+)](https://lightmysky.com/learn/mathematics/equivalent-fractions-age-9-mt_ebPelt-qAl)
- [Describing Rules & Patterns](https://lightmysky.com/learn/learning-to-learn/describing-rules-and-patterns-mt_hbe_kdE_7C)

## Opens up

- [Generalising with repeated reasoning](https://lightmysky.com/learn/mathematics/generalising-with-repeated-reasoning-mt_iyovKgZC1q)