How to Solve the Helen and Ivan Coin Problem: A Parent's Teaching Guide

If your child has encountered the Helen and Ivan coin question — or one like it — and you want to guide them through the solution without simply giving away the answer, this guide is for you. It explains not just the solution steps but the teaching moves that help a child internalise the reasoning, so they can apply the same thinking to unfamiliar problems in the future.

The goal is not to teach one question. The goal is to teach a way of thinking about comparison problems that transfers to any PSLE question with a similar structure.

To solve the Helen and Ivan coin problem, a child must recognise that when two people hold the same total number of coins but in different proportions of heavy and light coins, the difference in mass and value depends only on how many heavy coins one person holds relative to the other — and this difference can be found by comparing totals rather than by computing individual coin counts.

Before You Start: Set Up the Right Mindset

When a child looks at a problem like Helen and Ivan and says "I don't know how to start," the parent's first move is critical. Do not tell them the method. Instead, ask three questions that orient the child toward the problem's structure:

Question 1: "What is the same for both people?" This directs the child to look for invariants — the fixed element that everything else can be compared against. In the Helen and Ivan problem, the total number of coins is the same for both. This is the anchor.

Question 2: "What is different, and what causes the difference?" This directs the child to identify the variable that drives the outcome. Helen has more 50-cent coins; Ivan has more 20-cent coins. The difference in mass and value must come from this coin-type shift — not from one person having more coins overall.

Question 3: "Can we express the difference in terms of one thing?" This directs the child toward proportional reasoning. Each time we imagine swapping one 20-cent coin in Ivan's collection for one 50-cent coin (to match Helen's), the mass increases by the weight difference between the two coin types, and the value increases by 30 cents. The entire difference between the two collections can be expressed in terms of how many such swaps there are.

These three questions are reusable across a wide range of PSLE comparison problems. They are worth teaching as a general framework, not just as a solution to one question.

Guided Solution: Part (a) — Finding the Mass Difference

Walk your child through these steps, asking the question at each step and letting them fill in the answer:

Step 1: "What is the weight difference between one 50-cent coin and one 20-cent coin?" Answer: 3 g (given in the question).

Step 2: "If Helen has one more 50-cent coin than Ivan — and one fewer 20-cent coin to keep the total the same — how much heavier is her collection?" Answer: 3 g heavier, because one light coin was replaced by one heavy coin.

Step 3: "So if Helen has 18 more 50-cent coins than Ivan, how much heavier is her collection?" Answer: 18 × 3 g = 54 g heavier.

Step 4: "The question tells you Helen's total mass is 1,134 g. What is Ivan's total mass?" Answer: 1,134 g − 54 g = 1,080 g, or 1.08 kg.

The child may ask at Step 3: "How do I know Helen has 18 more 50-cent coins?" This number comes from additional information in the question — the total number of coins each person holds and the specific proportion — that allows the exact difference to be calculated. If the child is stuck on finding this number, walk them through using the total coin count and the total mass to set up the relationship, showing that the number of extra heavy coins is what makes the mass difference possible.

Guided Solution: Part (b) — Who Has More Money?

Once the mass difference is understood, the value comparison follows the same logic:

Step 1: "Each time Helen has a 50-cent coin instead of a 20-cent coin, how much more money does she have?" Answer: 30 cents more (50 − 20).

Step 2: "Helen has 18 more 50-cent coins. How much more money does she have in total?" Answer: 18 × 30 cents = 540 cents, or $5.40 more.

Step 3: "So who has more money?" Answer: Helen has more money than Ivan.

At this point, reinforce the key insight: the child did not need to know exactly how many coins of each type either person held. The comparative judgment was reachable through reasoning alone, using the relationship between the two collections. This is the meta-lesson the question is designed to teach.

If Your Child Gets Stuck: Diagnosis and Intervention

Different types of stuck-ness point to different underlying gaps:

If the child cannot identify what is the same for both people: The gap is in reading comprehension, not mathematics. Practise reading word problems aloud and identifying invariants — "what stays the same" — before any calculation. Use simpler comparison problems to build this habit before returning to multi-step ones.

If the child tries to set up equations and gets lost: The gap is in strategic flexibility. The child is defaulting to algebra because it is the only tool they trust, even when it is the wrong tool. Provide problems that cannot be solved by standard algebraic methods — comparison problems, working-backwards problems — to force the development of alternative strategies.

If the child gets the number right but cannot explain why: The gap is in mathematical communication, which is explicitly assessed in the PSLE. Practise asking "explain your answer" after every problem, not just the ones that ask for it. The explanation does not need to be elegantly written, but it must connect the reasoning steps to the conclusion.

Summary

Solving the Helen and Ivan coin problem means recognising that a comparison between two collections with the same total count can be expressed entirely in terms of how the coin types differ, without needing to know the exact composition of either collection. Teaching this to a child is most effective when parents ask orienting questions rather than providing the method, guide the child through each reasoning step rather than demonstrating the solution, and diagnose the specific gap when the child gets stuck rather than repeating the same explanation.

FAQ

What if my child insists on using algebra for every problem?

Algebra is a valid tool, but it is not always the best tool. When a child defaults to algebra for a problem where comparative reasoning is more efficient, they are not demonstrating mathematical sophistication — they are demonstrating a limited toolkit. Provide practice with problems where algebra is explicitly the wrong approach (underdetermined systems, comparison problems) to develop strategic flexibility. The goal is not to replace algebra but to add other tools alongside it.

How do I know if my child understands the reasoning or is just repeating the steps?

Ask the child to solve a structurally similar problem with different numbers and objects — for example, two people with different mixes of large and small bottles of water. If the child can transfer the comparative reasoning to the new scenario, they understand the method. If they revert to trying to calculate individual quantities, they have memorised the steps for Helen and Ivan without internalising the comparative reasoning framework.