How to Tackle a Mixed Mean Mass Problem

If you’re preparing for your GCSE Maths exam, you might come across questions involving ratios, fractions, and averages all in one go. The problem in the image above is a perfect example of how these concepts can be combined in a real exam question.

Understanding the Problem

You’re told there are 100 counters in a bag, and all are either red, green, or blue. The ratio of red to green counters is 4:1, and a quarter of the counters are blue. You’re given the mean mass of each colour of counter and asked to show that the mean mass of all 100 counters is less than 5 grams.

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Step 1: Work Out the Number of Each Colour

First, calculate how many blue counters there are:

• One quarter of 100 is \frac{1}{4} \times 100 = 25 blue counters.

That leaves 100 - 25 = 75 counters that are either red or green.

The red to green ratio is 4:1, so for every 4 red counters, there is 1 green counter.

Let the number of red counters be 4x and green counters be x.

So, 4x + x = 75 \implies 5x = 75 \implies x = 15

Therefore:

• Red counters = x = 15
• Green counters = x = 15
• Blue counters = 25

Step 2: Find the Total Mass

Now, use the mean mass for each colour:

• Total mass of red counters = 60 \times 6 = 360 grams
• Total mass of green counters = 15 \times 2 = 30 grams
• Total mass of blue counters = 25 \times 4 = 100 grams

Add them together for the total mass:
360 + 30 + 100 = 490 grams

Step 3: Calculate the Mean Mass

Mean mass of all 100 counters:
\frac{490}{100} = 4.9 grams

Step 4: Answer the Question

Since 4.9 < 5, you’ve shown that the mean mass of all 100 counters is less than 5 grams.

Key Takeaways

This problem brings together several important GCSE Maths skills: working with ratios, fractions, and averages. When faced with a question like this, break it down into manageable steps: find the quantities, calculate totals, and work out the average.

If you need more help with problems like this, a GCSE Maths Online Tutor can provide the guidance and practice you need to master these topics and boost your confidence for exam day.

Good luck with your revision!