If Gold = 5 and Silver = 15, how many times the number of Gold is Silver?

Three times. 'How many times' is a multiplicative comparison: it asks how many copies of the smaller value fit into the larger, which is division. 15 ÷ 5 = 3. The common wrong answer 10 is the difference 15 − 5 — that answers 'how many more', a different question. Subtracting gives how many more; dividing gives how many times.

How do you read the value of a bar off a bar chart?

Read the scale first. The value is the gridline count multiplied by the scale step, not the number of gridlines on its own. If the axis goes up in steps of 4 and a bar reaches the 3rd gridline, its value is 3 × 4 = 12, not 3. The width of the bar carries no information — only the height against the scale does.

How do you count a half symbol on a pictogram?

A part-symbol is that fraction of the key's value, not one whole item. If the key is ● = 8 books, a half symbol ◖ is half of 8 = 4 books. So a row of ● ● ◖ is 8 + 8 + 4 = 20 books. Counting the half as a whole 8 would give 24, which over-counts the part.

GCSE Maths Foundation · AQA · Statistical diagrams

Reading bar charts and pictograms: why the value comes from the scale and the key, not the appearance

Chart questions trip students who read the picture instead of the scale. Asked how many times bigger Silver (15) is than Gold (5), they answer 10 — the gap, not $15 \div 5 = 3$ . Asked the value of a bar at the 3rd gridline on an axis stepping in 4s, they answer 3, not $3 \times 4 = 12$ . Asked about a half symbol on a pictogram of $\bullet = 8$ , they count it as a whole 8, not $4$ .

The thirty-second fix: the value always comes from the scale and the key, never the appearance. ‘How many times’ means divide; a bar’s value is its height times the scale step; a part-symbol is a fraction of the key’s value.

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How to spot it in your own work

You answered “how many times” with the difference — writing $10$ for Silver vs Gold instead of $15 \div 5 = 3$ .
You read a bar’s value as the number of gridlines — answering $3$ for the 3rd gridline rather than $3 \times \text{scale step}$ .
You judged a bar by its width instead of its height against the scale.
You counted a pictogram part-symbol as one whole item — treating a half symbol on $\bullet = 8$ as $8$ rather than $4$ .

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Jun22 P3 Q7a shape — compare two bars):

A bar chart of 2008 Olympic medals shows Gold = 5 and Silver = 15. Silver is how many times the number of Gold?

The misconception answer is $10$ — the difference $15 - 5$ . But “how many times” asks how many copies of Gold fit into Silver.

Three 5s make 15, so Silver is three times Gold: $15 \div 5 = 3$ .

Why students fall for this

A chart hides its scale. The drawing shows sizes; the scale and the key carry the actual values. Because the picture is right there in front of the student, the conversion step — multiply by the scale step, or by what the key says — never happens, and the size of the drawing is read off as the answer.

“How many times” is the same error in words. It feels like a comparison, and subtraction is the comparison the student reaches for, so the difference gets written down. But “how many times” is multiplicative — it asks how many copies fit in — so it is division, not subtraction.

AQA Foundation papers exploit every face of this: comparing two bars with “how many times” (Jun22 P3 Q7a), critiquing a claim that a wider bar is bigger (Nov24 P1 Q8), reading a pictogram with a part-symbol (Nov24 P2 Q8), and part-symbol counting in context (Jun24 P2 Q5b).

The fix: The value is on the scale and the key: divide for 'how many times', multiply by the step or the key

How many times: divide the larger value by the smaller. For Gold = 5 and Silver = 15: $15 \div 5 = 3$ times. The difference $10$ answers “how many more”, not “how many times”.

Bar value: gridline count × scale step. On an axis stepping in 4s, the 3rd gridline is $3 \times 4 = 12$ . The width of the bar carries no value — only the height against the scale.

Pictogram: a symbol is worth what the key says. On $\bullet = 8$ , a full symbol is 8 and a half symbol is $8 \div 2 = 4$ . A row of $\bullet\, \bullet\, \text{(half)}$ is $8 + 8 + 4 = 20$ .

Worked example

A pictogram uses $\bullet = 8$ books. A row shows two full symbols and one half symbol. How many books does the row represent?

Read the key. One full symbol is worth 8 books.
$\bullet = 8$
Value the part-symbol as a fraction of the key.
$\tfrac{1}{2} \times 8 = 4$
Add the row.
$8 + 8 + 4 = 20$

The trap answer $24$ counts the half symbol as a whole 8. A part-symbol is a fraction of the key’s value, never one whole item.

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Common questions

What is the difference between “how many more” and “how many times”?: “How many more” is subtraction — Silver has $15 - 5 = 10$ more medals than Gold. “How many times” is division — Silver is $15 \div 5 = 3$ times Gold. They are different questions with different operations.
A bar chart’s axis goes up in 20s and a bar reaches the 7th gridline. What value does it show?: Multiply the gridline count by the scale step: $7 \times 20 = 140$ . The number $7$ is just how many gridlines the bar reaches — it is not the value until you scale it.
Does a wider bar show a bigger value?: No. The value of a bar is its height read off the scale, not its width. A wide, short bar can show less than a narrow, tall one. Width is only how the bar is drawn.
How many symbols does a row need for 12 people if the key is ● = 8?: Divide the frequency by the key: $12 \div 8 = 1.5$ symbols — one full symbol and one half symbol, because each symbol is worth 8 people.

Related misconceptions

Pie chart angles and frequency: why the angle is out of 360, not the number of peopleThe neighbouring statistical-diagram misconception — reading a number straight off the chart instead of converting through its hidden scale.
Computing mean and median: why you must sort before finding the middleAnother statistics recipe with a hidden structural step — the same 'apply the method, do not read the visible number' habit that chart-reading demands.
Interpreting range and spread: why the range is a single number, not an intervalA neighbouring statistics misconception where students read a value off the surface instead of computing the quantity the question asks for.

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