Exercises — Bar charts, pictograms, pie charts — drawing and reading

Most common = tallest bar = Cats (5 cm). You do not even need arithmetic to see this — the whole point of a bar chart is that the tallest bar wins. Dogs: height $\times$ scale $=3\times 4=12$ dogs. Why multiply? Each centimetre is a "package" of 4 pets, and there are 3 such packages.

Count symbols $\times$ value per symbol $=4\times 5=20$ apples. The icons are not the answer — the key tells you each icon is worth 5.

A full circle is $360^{\circ }$. Half of it is $\frac{1}{2}\times 360^{\circ }=180^{\circ }$, representing the fraction $\frac{1}{2}$ of the data. A straight line across the circle is a $180^{\circ }$ sector — look at the mint slice in the figure below.

Bar height $=\frac{\text{value}}{\text{scale factor}}$ (dividing converts book-units into drawing-cm):

• Monday: $24÷6=4$ cm
• Tuesday: $30÷6=5$ cm
• Wednesday: $18÷6=3$ cm

Number of symbols $=\frac{\text{data value}}{\text{value per symbol}}=\frac{20}{8}=2.5$ symbols → draw 2 full 🚌 and a half 🚌. The half-symbol carries the leftover $20-16=4$ buses, which is exactly half of 8.

Total $T=25+20+15=60$. Use $\theta =\frac{v}{T}\times 360^{\circ }$:

• Tea: $\frac{25}{60}\times 360^{\circ }=25\times 6=150^{\circ }$
• Coffee: $\frac{20}{60}\times 360^{\circ }=20\times 6=120^{\circ }$
• Juice: $\frac{15}{60}\times 360^{\circ }=15\times 6=90^{\circ }$ Check: $150+120+90=360^{\circ }$ ✓ — the slices fill the whole circle.

Handy shortcut: since $T=60$, each person is worth $\frac{360}{60}=6^{\circ }$, so you can just multiply each count by 6.

Run the formula backwards: $\text{value}=\frac{\theta }{360^{\circ }}\times T=\frac{54}{360}\times 240$. Simplify the fraction first: $\frac{54}{360}=\frac{3}{20}$. $\frac{3}{20}\times 240=3\times 12=36$ students. Why backwards? We have the angle and want the value, so we invert the "value → angle" machine.

Let one symbol be worth $x$ apples. Shop A $=4x$, Shop B $=6x$. The difference is $6x-4x=2x$, and we're told this equals 30: $2x=30⟹x=15\text{ apples per symbol}.$ Check: Shop A $=60$, Shop B $=90$, difference $=30$ ✓.

The ratio of angle to value is constant across the whole chart (both are proportional to the fraction). So set up a proportion (see Ratio and proportion): $\frac{120^{\circ }}{8}=\frac{360^{\circ }}{T}⟹T=\frac{360\times 8}{120}=24\text{ items total}.$ Now the $75^{\circ }$ sector: $\text{value}=\frac{75}{360}\times 24=\frac{75\times 24}{360}=5$ items. The key insight: degrees per item $=\frac{120}{8}=15^{\circ }$ per item is fixed, so $75^{\circ }÷15=5$ items — same answer, faster.

Total: $T=12+9+6+3=30$. Degrees per student $=\frac{360}{30}=12^{\circ }$.

• Pizza: $12\times 12^{\circ }=144^{\circ }$
• Salad: $9\times 12^{\circ }=108^{\circ }$
• Sandwich: $6\times 12^{\circ }=72^{\circ }$
• Soup: $3\times 12^{\circ }=36^{\circ }$ Check: $144+108+72+36=360^{\circ }$ ✓ — a complete, honest circle.

Percentage $=\frac{12}{30}\times 100\%=40\%$. Turn percent into angle: $1\%$ of the circle is $\frac{360^{\circ }}{100}=3.6^{\circ }$, so $40\%\times 3.6^{\circ }=144^{\circ }$. Reassuringly, this matches the $144^{\circ }$ from L4.1 — percentage and direct fraction are two roads to the same slice. See Fractions and percentages.

(a) Walk angle: angles fill the circle, so ${\theta }_{\text{walk}}=360^{\circ }-(150+120+60{)}^{\circ }=360^{\circ }-330^{\circ }=30^{\circ }.$

(b) People per degree $=\frac{720}{360}=2$ people per degree. Multiply each angle by 2:

• Car: $150\times 2=300$ people
• Bus: $120\times 2=240$ people
• Cycle: $60\times 2=120$ people
• Walk: $30\times 2=60$ people Check: $300+240+120+60=720$ ✓.

(c) Redrawn pie after +90 cyclists: the total changes, so all angles must be recomputed — this is the whole trap of the question.

• New cycle count $=120+90=210$.
• New total $T^{\prime }=720+90=810$.
• New cycle angle $=\frac{210}{810}\times 360^{\circ }=\frac{210\times 360}{810}=\frac{75600}{810}=93.33^{\circ }$ (that is $93{\frac{1}{3}}^{\circ }$).

(d) With many categories a pie becomes a cluttered ring of thin slivers that are hard to compare by eye; a bar chart lines them up on a common scale so exact values and rankings stay readable. (See the "when to use each chart" guide in the parent note.)

Pie angle from data ::: $\theta =\frac{v}{T}\times 360^{\circ }$ Value from pie angle ::: $\text{value}=\frac{\theta }{360^{\circ }}\times T$ Bar/pictogram value ::: (units or symbols) $\times$ (scale or value per symbol) After data changes in a pie, do what first? ::: recompute the total, then every angle