Percentages, ratios and averages inside DI

DI questions are built from a handful of arithmetic ideas: percentage, ratio, average and percentage change. None of them is hard on its own. The trouble is that DI puts them inside a table under time pressure, and a few specific slips catch people repeatedly. Examiners know this and put the wrong answer from each slip among the options.

This guide lists the seven traps that cost the most marks in bank PO DI, explains why the correct method works, then puts them together in one worked set so you can see them in action.

Trap 1: averaging percentages

When groups are of different sizes, the overall percentage is total ÷ total, not the average of the percentages.

School X has 500 students with an 80% pass rate. School Y has 300 students with a 60% pass rate.

• Wrong: (80 + 60) ÷ 2 = 70%.
• Right: passes = 400 + 180 = 580 out of 800, so 72.5%.

Why. The simple average treats both schools as equal. X is bigger, so its rate should pull the overall figure towards 80%. The overall rate always lies between the group rates and closer to the bigger group's.

Trap 2: percentage points versus percent

An interest rate rises from 6% to 7.5%.

• The rise is 1.5 percentage points (7.5 − 6).
• The percentage increase is 1.5 ÷ 6 × 100 = 25%.

Read which one the question asks for. "By how much did the share rise" with options in points is the first; "by what percent" is the second.

Trap 3: the base after "than"

If A = 150 and B = 120:

• A is (150 − 120) ÷ 120 × 100 = 25% more than B.
• B is (150 − 120) ÷ 150 × 100 = 20% less than A.

The base is always the quantity after "than". The two answers differ because the same gap of 30 is measured against different bases.

Trap 4: ratios the wrong way round

"The ratio of A to B" is A : B. Write the quantities in the order the question names them before simplifying. If the options include both 7 : 12 and 12 : 7, one of them is the trap.

Trap 5: averages from totals

If the average of 6 values is 40 and a value of 70 is added, the new average is (6 × 40 + 70) ÷ 7 = 310 ÷ 7 ≈ 44.3, not (40 + 70) ÷ 2 = 55.

Why. An average is a total divided by a count. When anything changes, rebuild the total. A useful form: new value = new average + (old count × rise in average).

Trap 6: successive changes are not added

A value rises 20% and then falls 20%. It does not return to where it started: 100 → 120 → 96, a net 4% fall.

Why. The second change works on a different base. The general rule for changes of a% and b% is a + b + ab/100, using minus signs for falls. Here: 20 − 20 + (20 × −20)/100 = −4.

Trap 7: reverse percentages

After a 12.5% rise, a figure is 540. The original is 540 ÷ 1.125 = 480, not 540 minus 12.5% of 540 (472.5).

Why. The 12.5% was calculated on the original value, so you must divide by 1.125 to undo it. Using fractions is faster: 12.5% = 1/8, so the new value is 9/8 of the old, and the old is 540 × 8/9 = 480.

The seven traps at a glance

A worked set with the traps built in

The table shows the number of customers at three branches and the percentage who use UPI and net banking. These are made-up figures for practice.

Helper figures: UPI users are P 480, Q 540, R 360. Net banking users are P 280, Q 480, R 150.

Q1. What percentage of all customers of the three branches use UPI?

• (480 + 540 + 360) ÷ 2,500 × 100 = 1,380 ÷ 2,500 × 100 = 55.2%.
• Trap option: (60 + 45 + 72) ÷ 3 = 59%.

Q2. By how many percentage points, and by what percent, is Q's UPI rate below P's?

• 60 − 45 = 15 percentage points.
• 15 ÷ 60 × 100 = 25% lower.

Q3. UPI users at P are what percentage more than UPI users at R? And R is what percentage less than P?

• (480 − 360) ÷ 360 × 100 = 33.33% more.
• (480 − 360) ÷ 480 × 100 = 25% less.

Q4. What is the ratio of net banking users at Q to net banking users at P?

• 480 : 280 = 12 : 7.

Q5. The average number of customers per branch is about 833. A new branch S opens and the average for the four branches becomes 900. How many customers does S have?

• New total = 4 × 900 = 3,600. S = 3,600 − 2,500 = 1,100.
• Check with the shortcut: rise in average = 900 − 833⅓ = 66⅔. S = 900 + 3 × 66⅔ = 1,100.

Practice set

1. Branch P has 250 customers, 40% of whom use UPI. Branch Q has 150 customers, 70% of whom use UPI. What percentage of all customers use UPI?
2. A rate rises from 8% to 9%. By how many percentage points, and by what percent?
3. A = 90 and B = 120. A is what percent less than B?
4. The average of 5 values is 30. A sixth value of 60 is added. What is the new average?
5. A branch's deposits rise 20% one year and fall 20% the next. What is the net change?
6. After a 12.5% rise, a branch's loan book is ₹540 crore. What was it before the rise?
7. From the worked set: what percentage of all customers use net banking?

Answers:

1. 51.25%. (100 + 105) ÷ 400 × 100. The simple average (55%) is the trap.
2. 1 percentage point; 12.5%. 1 ÷ 8 × 100 = 12.5.
3. 25%. 30 ÷ 120 × 100. The base is B, the quantity after "than".
4. 35. (150 + 60) ÷ 6 = 210 ÷ 6.
5. A 4% fall. 20 − 20 − 400/100 = −4.
6. ₹480 crore. 540 × 8/9.
7. 36.4%. (280 + 480 + 150) ÷ 2,500 × 100 = 910 ÷ 2,500 × 100.

What to do next

• Make a one-page sheet of the seven traps and read it before every DI mock for two weeks.
• After each mock, tag every DI error with its trap number. Most people find that two or three traps cause most of their errors.
• Apply these to full sets: table DI, pie chart DI and caselet DI.