Tables, Bar Graphs, Line Graphs & Pie Charts

A table or chart organises data so you can compare quantities. Read its labels and units before calculating.

Write the denominator next to every percentage. Convert different charts to comparable quantities before ranking them.

A table shows 12 books borrowed on Monday and 18 on Tuesday. How many were borrowed in total? 1. Both entries measure books borrowed. 2. Add the entries: 12 + 18. 3. The total is 30 books.

The vertical axis measures books, and the two equally spaced bars represent the same unit on different days. Read 12 and 18, then add them to obtain 30. Their difference is 6 books. Read labels and scales before adding values from different charts.

Region A sells 200 units, 30% premium; B sells 300 units, 20% premium. Both sell 60 premium units. Next year A's total rises 25% and premium share falls to 24%: 250 x 0.24 = 60. B's total falls 10% and share rises to 25%: 270 x 0.25 = 67.5 in the continuous sales-volume model. B now has the larger premium volume despite the lower total. If units are indivisible objects, the stated percentages would need rounding or different data.

A chart showing percentages cannot be treated as raw counts without knowing the total. Percentage points and percentage changes differ. A rise from 20% to 25% is 5 points but a 25% relative increase.

1. Find the row or bar named in the question. 2. Check its unit. 3. Write down only the numbers you need. 4. Decide whether to add, subtract, compare, or find a percentage. 5. Check that your answer uses the same unit.

A table shows 20 books borrowed on Monday and 30 on Tuesday. The increase is 30 - 20 = 10 books. The percentage increase uses Monday as the starting value: 10 / 20 x 100 = 50%. It is not 10 / 30.

A bigger slice can still represent a smaller number if two pie charts have different totals. Always check the total before comparing percentages across charts.

Mixed displays combine units or measures. Match each series to its axis and time period; never divide values just because they are adjacent.

Revenue is 900 thousand rupees and volume is 300 units. Average revenue per unit is 3000 rupees. The thousand multiplier must be applied before interpreting the answer.

Ratios compare quantities in consistent units. Combine ratios only after matching the common quantity.

If A:B = 2:3 and B:C = 4:5, multiply the first ratio by 4 and the second by 3. Then A:B:C = 8:12:15.

Estimate the scale first, simplify fractions and calculate only what the question needs. Round only when the answer gap permits it.

For 49.8% of 802, half of 802 is 401, so the answer is close to 399.4. An option near 200 is impossible; close options still require the exact calculation.

Write the denominator next to every percentage. Convert different charts to comparable quantities before ranking them.

Region A sells 200 units, 30% premium; B sells 300 units, 20% premium. Both sell 60 premium units. Next year A's total rises 25% and premium share falls to 24%: 250 x 0.24 = 60. B's total falls 10% and share rises to 25%: 270 x 0.25 = 67.5 in the continuous sales-volume model. B now has the larger premium volume despite the lower total. If units are indivisible objects, the stated percentages would need rounding or different data.

Percentage points and percentage changes differ. A rise from 20% to 25% is 5 points but a 25% relative increase.

Read axis labels, units, scale and series keys before comparing values. Line graphs connect an ordered series; grouped bars compare categories; stacked bars show contributions to totals. A 100% stacked chart shows shares, so equal-height totals need not represent equal actual amounts.

Sales of A and B are 80 and 120 in year one, and 100 and 150 in year two. Both grow by 25%: (100−80)/80 and (150−120)/120. Totals rise from 200 to 250, and A remains 40% of each total. A line between observed years does not establish every intervening value without an interpolation assumption.

For a stacked segment, subtract its bottom boundary from its top boundary. Use absolute totals before comparing the monetary size of percentage segments in different bars. If an axis is truncated, compute from labelled values rather than apparent bar lengths. A dual-axis chart may use different units for its series.

Equal numerical increases are not equal percentage increases when the starting values differ. A chart's visual slope changes with axis scaling. Correlation alone does not prove causation, and a missing original graph cannot be reconstructed from scattered axis labels.

A full pie represents one total and 360°. A sector of angle θ degrees represents θ/360 of that total. Sector percentages sum to 100% apart from stated rounding.

A construction budget is 600000. Sector angles are timber 36°, labour 90°, cement 72°, bricks 54°, steel 54° and supervision 54°, summing to 360°. Cement costs (72/360)×600000=120000. Labour exceeds steel by (90−54)/360×600000=60000, which is 10% of the total budget. It exceeds steel's own cost by 36/54×100≈66.67%; that is a different base. Cement, steel and supervision together span 72+54+54=180°, so they account for 50% of the total. Timber costs (36/360)×600000=60000.

The cement amount is 120000; an option label inconsistent with that calculation is not an answer key. Comparing two pies requires their totals: 20% of 1500 is 300, greater than 30% of 800, which is 240.

Every table cell belongs to a row, column, unit and population. Distinguish totals, averages and percentages. When only a group's size and minimum/maximum are known, many individual values remain undetermined.

Five respondents have minimum age 34 and maximum age 49. At least one is under 40, because a 34-year-old must exist, and at most four are under 40, because a 49-year-old must exist. Four can also be older than 35, while the minimum forces at least one not older than 35. These bounds need not be exact counts. For an inclusive interval [35,40], a group whose minimum and maximum are both below 35 has no members inside; so does a group whose endpoints are both above 40. If its minimum is 34 and maximum 49, no member is forced inside [35,40]. If its minimum is 35 and maximum 40, all members are inside. A minimum and maximum on opposite sides of the interval do not guarantee an interior observation.

Derive a lower and upper count for each group, respecting whether thresholds are strict or inclusive. Add group bounds only when groups do not overlap. Percentage = qualifying count / relevant total × 100. One million equals ten lakh; apply the multiplier before comparing values.

Do not assume uniform age distribution or fill in every missing age. Approximation can separate widely spaced options, but close options require enough precision to establish the comparison.