Data Interpretation: Charts, Growth Rates & Percentage Points

A pie is a circle, so 360° = 100% and 1% = 3.6°. The conversions worth memorising: 36° = 10%, 45° = 12.5%, 54° = 15%, 72° = 20%, 90° = 25%, 120° = 33⅓%, 180° = 50%. Value of a slice = (degrees ÷ 360) × total. Worked: total expenditure ₹7,200 crore, the education slice subtends 54°. Share = 54/360 = 0.15 = 15%. Value = 0.15 × 7,200 = ₹1,080 crore. In reverse: a slice worth ₹1,800 crore is 1,800/7,200 = 25% = 90°. Never multiply the total by the degrees as if they were a percentage — 54% of 7,200 is 3,888, and that number will be among the options.

A percentage change divides the change by the old value; a percentage-point change simply subtracts two percentages. Worked: a bank's interest rate moves from 6% to 7.5%. Percentage-point change = 7.5 − 6 = 1.5 percentage points. Percentage change = 1.5/6 = 25%. Both are correct descriptions and they differ by a factor of sixteen. The paper will print "rose by 1.5%" and "rose by 25%" as separate options; only the second is a percentage change, and "1.5%" is not what happened to the rate — the rate rose by a quarter of itself.

A ratio change is a third thing again. If the ratio of boys to girls moves from 2 : 3 to 3 : 4, the ratio has gone from 0.667 to 0.750 — a rise of 0.083/0.667 = 12.5% in the ratio. This says nothing about how many boys there are: 200 boys and 300 girls becoming 150 boys and 200 girls gives exactly that ratio change with the number of boys falling. Ratios, shares and percentages all describe relationships; only an absolute figure describes a quantity, and every DI question is secretly asking which of the two it wants.

1. Name the base. Before dividing, write down which value is the "old" one. 40 → 50 is +25%; 50 → 40 is −20%. The base is always the value you are changing from.
2. Ask whether the question wants points or percent. "By how many percentage points did the share rise?" is a subtraction. "By what percent did the share rise?" is a division. Read the unit in the options — "pp" or "%" — before touching the numbers.
3. Ask whether the question wants a quantity or a relationship. "Whose sales grew most?" needs absolute numbers; "whose share grew most?" needs only the shares. Answering one when the other was asked is the standard wrong option.

Successive growth multiplies; it does not add. Growth of a% then b% gives a total of a + b + ab/100 percent. Worked: revenue grows 20% in year one and 25% in year two. Multiplier = 1.20 × 1.25 = 1.50, so total growth = 50%, not 45%. By the formula: 20 + 25 + (20 × 25)/100 = 45 + 5 = 50 ✓. The same rule with a fall: +20% then −20% gives 1.20 × 0.80 = 0.96, a net fall of 4% — a rise and an equal-percentage fall never cancel, because the fall is taken on a bigger base.

CAGR (compound annual growth rate) is the single rate that, applied every year, takes the initial value to the final one: CAGR = (Final ÷ Initial)^(1/n) − 1, where n is the number of years elapsed — 2019 to 2021 is n = 2, not 3. Worked: a market grows from ₹600 crore in 2019 to ₹800 crore in 2021. Ratio = 800/600 = 1.3333. We need the square root: 1.15² = 1.3225 (too small), 1.16² = 1.3456 (too big), so CAGR ≈ 15.5% (exact 15.47%). The quick approximation: total growth ÷ n is always an upper bound — here 33.3%/2 = 16.7% — and the true CAGR is a little below it, by more the larger the growth. Test a candidate by multiplying back: 1.155² ≈ 1.334 ✓.

1. Write the ratio Final/Initial and count the years elapsed. Half the CAGR errors in the paper are an n that is off by one.
2. Look for doublings. A ratio of 2 is one doubling, 4 is two, 8 is three. If 800 became 3,200 in 10 years, it doubled twice, so the doubling time is 5 years.
3. Apply the Rule of 72: doubling time × rate ≈ 72. A doubling time of 5 years means a rate of about 72/5 ≈ 14.4%; a rate of 9% means a doubling in about 8 years.
4. For anything else, take total growth ÷ n and shade it down. Then multiply your candidate back up to check it against the options before committing.

Worked: a town's population grows from 200 thousand to 800 thousand in 12 years; find the CAGR. Ratio = 4 = two doublings, so one doubling took 6 years. Rule of 72: rate ≈ 72/6 = 12%. (Exact: 4^(1/12) = 2^(1/6) = 1.1225, so 12.25%.) The wrong options will be 300%/12 = 25% (treating the total growth as if it were simple interest) and 400%/12 = 33% (dividing the ratio rather than the growth). Both are far off because compounding over twelve years is not remotely linear.

Market share = company sales ÷ market sales, so a change in share depends on two growth rates: new share ÷ old share = (1 + company growth) ÷ (1 + market growth). Worked: a company's sales grow 10% while the market grows 25%. Share multiplier = 1.10/1.25 = 0.88, so the share falls by 12% — even though sales rose. If the old share was 20%, the new share is 20 × 0.88 = 17.6%, a fall of 2.4 percentage points. The wrong option is 25 − 10 = "fell by 15%": subtracting growth rates is base-year mixing in another costume.

Within one pie or one year, the ratio of two companies' sales equals the ratio of their shares, and the total cancels: shares of 30% and 24% mean the first sells 30/24 = 1.25 times the second, whatever the market size. Across two years the totals do not cancel. A slice of 30% of ₹400 crore is ₹120 crore; a slice of 24% of ₹600 crore is ₹144 crore. The share fell 6 points and the amount rose 20%. Every "which segment grew fastest?" question on a pair of pies is this calculation, once per segment, and it cannot be done from the percentages alone.

• Round to two significant figures before dividing. 4,827 ÷ 26,140 becomes 48 ÷ 261 ≈ 0.184, so 18.4%; the exact value is 18.47%. Two figures are enough when options are more than two points apart, which in CAT they almost always are.
• Compare two fractions by cross-multiplication, never by dividing both. Is 17/23 bigger than 22/29? 17 × 29 = 493 and 22 × 23 = 506; the second product belongs to 22/29, so 22/29 is larger. This is two multiplications instead of two long divisions.
• Build percentages from 10% and 1%. 23% of 640: 10% is 64, so 20% is 128; 1% is 6.4, so 3% is 19.2; total 147.2. Faster and safer than multiplying 640 × 0.23 under time pressure.
• Use the fraction anchors to see a percentage without dividing. 91 out of 637: 637 = 7 × 91, so the answer is exactly 1/7 = 14.3%. 85 out of 680: 680 = 8 × 85, so 12.5%. Setters build tables from these fractions because the questions are meant to be quick.

The table below is the kind of stimulus a DI set carries: four companies, three years, revenue in ₹ crore, with a totals row. Four questions follow, each answered with every line of arithmetic shown. Cover the answers and attempt each one first; the wrong turns are noted so that you can see which trap each question was built on.

Q1. By how many percentage points did Alpha's market share change from 2019 to 2021, and by what percent? Share 2019 = 240/600 = 0.40 = 40%. Share 2021 = 360/800 = 0.45 = 45%. Percentage-point change = 45 − 40 = +5 pp. Percentage change in share = 5/40 = +12.5%. Note that Alpha's revenue grew 120/240 = 50% — a third number, and the one a hurried candidate reports for "how did Alpha do?". Three different correct figures, three different questions.

Q2. Which company grew fastest from 2019 to 2021? Alpha: 360/240 = 1.50 → +50%. Beta: 200/160 = 1.25 → +25%. Gamma: 112/120 = 0.933 → −6.7%. Delta: 128/80 = 1.60 → +60%. Delta grew fastest, though its absolute increase (+48) is the smallest of the three risers and Alpha's (+120) the largest. "Fastest" is a relative word and always means percentage; "most" means absolute. Also notice that Delta fell in 2020 (80 → 64) before rising — a growth question on end-points ignores the path.

Q3. Alpha's year-on-year growth rates, and the CAGR of the total market. Alpha 2019 → 2020: 288/240 = 1.20 → +20%. 2020 → 2021: 360/288 = 1.25 → +25%. Combined: 1.20 × 1.25 = 1.50 → +50%, which matches Q2 (and is not 20 + 25 = 45%). Total market: 600 → 800 over n = 2 years, ratio 1.3333; √1.3333 ≈ 1.155, so CAGR ≈ 15.5%. The average of the two yearly growths of the total, (6.67% + 25%)/2 = 15.8%, is close but not equal — averaging growth rates is another approximation, and always an over-estimate.

Q4. Drawn as a 100% stacked bar, did Beta's segment grow or shrink from 2019 to 2021? Beta's share 2019 = 160/600 = 26.67%; 2021 = 200/800 = 25%. The segment shrank by 1.67 percentage points while Beta's revenue rose 25% — because the bar (the market) grew 33.3%, faster than Beta. Gamma in 2020 shows the reverse trap: 96/640 = 15% against 120/600 = 20% in 2019, a 5-point fall in share that understates a 20% fall in revenue. A stacked-bar segment tells you about the company relative to the market; only the totals row tells you about the company.

• Averages of percentages are not the percentage of the total. Division P grows 10% on a base of 100 and Division Q grows 30% on a base of 300: total growth is (10 + 90)/400 = 25%, not the average 20%. A percentage must be weighted by its base before it can be combined.
• Absolute versus relative is decided by one word in the question. "Largest increase" is a subtraction; "highest growth" is a division. In the table above they name different companies (Alpha and Delta), and a set will always be built so that they do.
• A segment and its bar move independently. Check the totals row before saying anything about a company from a stacked bar or a pie. If the total is not given, the honest answer about amounts is "cannot be determined", and CAT does print that option.