A relative frequency records how often an event occurred among the observed trials: event count divided by trial count. Eight successes in ten trials gives 8/10 = 0.8. It is a summary of those ten trials, not a promise that the next ten will contain exactly eight successes.

Define success consistently. In a spinner experiment, ‘lands on red’ is a different event from ‘lands on a warm colour’. Counts cannot be combined sensibly if the word success silently changes between batches. Write the event and the total beside each count before comparing rates.

An ordinary average of the percentages treats the batches as two equal units. It answers a batch-weighted question: what is the mean of the two reported rates? It does not answer what fraction of all individual trials were successes when batch sizes differ.

In the example, the first percentage represents ten trials and the second represents ninety. The second batch supplies nine times as many trial observations. Giving them equal influence discards that difference. Return to 8 and 18 successes, and 10 and 90 trials, to preserve what each denominator represents.

Add the event counts: 8 + 18 = 26. Add the trial counts: 10 + 90 = 100. The pooled relative frequency is 26/100 = 0.26 = 26%. This is an exact description of the combined recorded data, assuming the counts and event labels are correct.

Check that successes do not exceed trials in either batch and that no trial is counted twice. Combining a first-ten-trials report with a later cumulative first-hundred-trials report would double-count the first ten. Establish whether batches are disjoint before adding their totals.

The same calculation can be written (10 × 0.8 + 90 × 0.2) ÷ 100. The weights are the trial counts. Multiplying each relative frequency by its denominator recovers its event count, so this form gives exactly the same result as pooling counts.

Use rates as decimals in this expression, or keep percentage notation consistently. Do not enter 80 and 0.2 together as if they share a scale. Original counts are often the clearest route because they keep the numerator and denominator visible and reduce unnecessary conversions.

If the batches have equal numbers of trials, the simple mean of their relative frequencies equals the pooled relative frequency. For 6 successes in 20 trials and 14 successes in 20 trials, the rates are 30% and 70%. Both methods give 20/40 = 50%.

With unequal batch sizes, the methods can still coincide if the rates happen to be equal. The dependable reason to choose a method is the requested quantity and the weights, not whether one example happened to agree. Decide whether the unit being averaged is a batch or an individual trial.

For positive batch sizes, the pooled rate is a weighted average and lies between the smallest and largest batch rates, inclusive. In the main example, 26% lies between 20% and 80% and is closer to 20%, reflecting the larger second batch.

This gives a useful arithmetic check. A pooled answer of 92% would be impossible for these two batches. However, being between the rates does not prove correctness: 50% also lies between them. Use this as a reasonableness test after calculating the counts, not as a replacement for that calculation.

Exact counts may not be recoverable. A report of 33% from 30 trials might be a rounded display of 10/30, but multiplying 0.33 by 30 gives 9.9, which is not a count. Use the original event totals whenever they are available.

If the task supplies rounded rates only, explain that a calculated pooled result may be approximate and follow the problem’s instructions. Do not quietly round fractional reconstructed counts and present them as known observations. The precision of the conclusion cannot exceed the information supplied.

A pooled proportion still describes all the recorded trials if the event is consistently defined. But its usefulness as an estimate for a future trial depends on the target situation. Different spinners, changed equipment or altered recording rules can make one combined rate conceal important differences.

For example, combining results from an easy setting and a difficult setting gives a rate for that particular mixture. It may not predict either setting well on its own. State which population or future process the estimate is meant to describe, and keep separate summaries when the conditions matter.

No. More observations under a suitable stable random model can improve the information available about a probability, but they do not determine the next outcome. A relative frequency of 26% does not mean success is scheduled once every four trials or that recent failures force a success.

Distinguish observation from model. The exact recorded proportion is 26/100. Treating 0.26 as the probability for future trials is a modelling choice that needs comparable conditions. The arithmetic can be exact while the prediction remains uncertain. A clear answer names both roles.

Question 1: batches record 3/5 and 21/35 successes; find the pooled rate. Question 2: batches record 9/10 and 9/30; find the pooled rate and the simple mean of the two rates. Question 3: batches record 4/20 and 12/20; compare the two methods.

Question 4: a report gives 8/10 first, then 26/100 cumulatively. Should the figures be added as separate batches? Answers: 1) 24/40 = 60%. 2) Pooled 18/40 = 45%; simple mean 60%. 3) Both give 40%. 4) No; the first ten are already included in the hundred.

Ask the student to label success counts, trial counts and the requested unit of comparison before calculating. A tutor can then change one batch size while keeping both rates fixed. The child should predict which way the pooled rate moves before seeing the new answer.

In a small-group 3-pax lesson, each learner can explain one choice independently: why counts are added, when equal weighting works, or why changed conditions limit prediction. Bring the original worksheet and working so the lesson targets a specific denominator decision rather than repeating a whole probability chapter.

Define the event. Check that batches are separate and that counts refer to the same event. Decide whether the question asks about all trials or the average batch. For all trials, add successes, add trials and divide. Retain exact counts until the final conversion or rounding step.

For parents considering Secondary 4 Mathematics tuition in Punggol, the useful checkpoint is, ‘What does the denominator count here?’ When that answer is clear, percentages become easier to combine and the final statement becomes easier to interpret responsibly.