Percentages: Part, Whole and Comparison Base

What is the percentage being taken of?

Two reports can show the same number of errors and describe very different situations. The missing information is the number of opportunities for an error. A percentage supplies that comparison only when its base is clear.

In this lesson you will identify the base, move between fractions, decimals and percentages, find an unknown part or whole, calculate a change from the correct starting value, and check a result by estimation. You will also learn when the information does not determine an answer.

You need multiplication, division, simple fractions and decimal place value. Quick readiness check: one-quarter of 20 is 5; 0.2 is 1/5; 1.4 km is 1400 m. If any of these is unclear, revise that operation before continuing.

1. Give 100% a name

“Percent” means “per hundred.” A rate of 7% is the ratio 7/100. It does not mean seven objects unless the whole happens to contain 100 objects.

Use three labels:

• B, the base or whole: the quantity being treated as 100%
• A, the compared amount or part: the quantity corresponding to the stated percentage
• p, the percentage number: for 7%, p is 7, and the decimal rate is p/100 = 0.07

The relationship is:

A = B × p/100

Multiplying a quantity by a dimensionless rate preserves its unit. If B is in litres, A is in litres. If you divide two lengths to find a percentage, put the lengths in the same unit first.

For an ordinary part of a fixed whole, A lies between 0 and B. A comparison can exceed 100%: a new output may be 125% of an earlier output. That means 1.25 times the baseline. It does not mean that a fixed group contains more distinct members than its total membership.

Teaching panel 1: three ways to express the same rate

Percent	Decimal rate	Fraction
100%	1	1
50%	0.5	1/2
25%	0.25	1/4
12.5%	0.125	1/8
1%	0.01	1/100
0.5%	0.005	1/200
125%	1.25	5/4

To convert a percentage to a decimal, divide its number by 100. To convert a decimal rate to a percentage, multiply by 100 and write the percent sign. These two operations reverse each other. In particular, 0.5 and 0.5% are different: 0.5 is 50%, while 0.5% is 0.005.

Coached check A. Write 0.045 as a percentage. Multiplying by 100 gives 4.5%, not 0.045%. Check backwards: 4.5/100 = 0.045.

2. Choose the unknown before choosing the operation

The same relationship can answer three different questions. Start by naming the known quantities rather than reaching for a remembered shortcut.

Teaching panel 2: the base–amount–rate ledger

What is missing?	Write first	Then calculate	Condition
Amount A	A = B × p/100	Multiply the base by the rate	The base and rate must be supplied
Percentage p	A/B = p/100	p = 100 × A/B	B must be nonzero
Whole B	A = B × p/100	B = A ÷ (p/100)	p must be nonzero

For the positive quantities in this lesson, a part-of-whole rate below 100% gives an amount smaller than the whole. When recovering that whole, the answer must therefore be larger than the part. This is a useful direction check, not a replacement for calculation.

Coached check B. Suppose 16 is 20% of an unknown whole. Since 20% is 1/5, the whole is five equal parts of 16, or 80. Multiplying 16 by 0.2 would give 3.2, which is one-fifth of the already-known part.

When a zero prevents division

• To find a percentage, the base cannot be zero. “12 as a percentage of 0” has no defined value under this ratio model.
• To recover a whole from a percentage, the rate cannot be zero. “0 is 0% of the whole” fits any whole; it does not prove that the whole is zero.
• A nonzero amount described as 0% of a finite whole is inconsistent with this model.
• A percentage alone gives a relative amount, not an absolute count. “60% attended” does not tell you how many attended unless the total is known.

Coached check C. A statement says that 40% of a batch passed inspection, but gives no batch size. You can say that the other 60% did not pass if the two categories are exhaustive and non-overlapping. You cannot give either count. A batch of 50 and a batch of 500 would produce different counts while satisfying the same rate.

3. A change has a starting base

For a change from an old value O to a new value N:

Change in amount = N − O

Percentage change = (N − O)/O × 100%, provided O ≠ 0

For the positive starting values considered here, a positive result describes an increase; a negative result describes a decrease. If the question asks for the percentage decrease, give its positive size: (O − N)/O × 100% when N < O.

The denominator is the starting value. Dividing by the new value asks a different comparison. Also distinguish the change from the new value expressed as a percentage of the old one: a new value that is 110% of the old value represents a 10% increase.

Teaching panel 3: follow the comparison arrow

Comparison	Numerator	Base
Change from old to new	New − old	Old
New as a percentage of old	New	Old
Old as a percentage of new	Old	New

Coached check D. A count moves from 100 to 110. The change is 10, so the increase is 10/100 × 100% = 10%. The new count is 110% of the old count. Both statements are correct, but they answer different questions.

If an old count of zero becomes a positive count, report the absolute increase. The percentage-change formula would divide by zero; neither a guessed finite percentage nor “infinity percent” is a valid result from this formula.

4. Estimate before calculating exactly

An estimate should help you reject a misplaced decimal or wrong operation. It must not be reported as an exact answer.

• Choose a nearby convenient percentage or base.
• State whether the result is approximate.
• When the choice depends on a narrow range, establish a bound instead of assuming that rounding will be harmless.
• Keep full values during a calculation. Round the final result only when the question asks for rounding, and follow the stated precision.

With a positive rate, multiplying a lower and an upper bound for the base preserves their order. For example, if 200 < B < 210, then 20 < 10% of B < 21. This works because 10% is positive. The same reasoning will justify the bound in worked example 6.

Coached check E. Is 4% of 300 closer to 12 or 120? One percent is 3, so four percent is 12. A result of 120 would be 40% of 300. The benchmark catches a factor-of-ten error.

5. Six worked examples

Worked example 1: find the amount

A workshop has 240 components. Exactly 17.5% are aluminium. How many components are aluminium?

Identify. The whole is 240 components and the rate is 17.5/100 = 0.175.

Calculate. A = 240 × 0.175 = 42 components.

Explain another route. Split 17.5% into 10% + 5% + 2.5%. The corresponding amounts are 24 + 12 + 6 = 42. Each part uses the same whole of 240.

Check. The result is below the whole and between 10% and 20% of it. Dividing 240 by 17.5 would not calculate a percentage of 240.

Worked example 2: find the rate

A sensor flags 63 readings out of 420. What percentage of the readings are flagged?

Identify. The compared amount is 63 readings. The base is all 420 readings.

Calculate. p = 100 × 63/420 = 100 × 3/20 = 15. Therefore 15% are flagged.

Check. Ten percent of 420 is 42 and five percent is 21; together they are 63. The units cancel in 63 readings / 420 readings. Reversing the fraction would compare all readings with the flagged subset and answer a different question.

Worked example 3: recover the whole

A library issues 27 reference books, equal to 12.5% of all books issued that morning. How many books were issued in total?

Identify. The part is 27, and 12.5% = 1/8 of the whole.

Calculate. B = 27 ÷ (1/8) = 27 × 8 = 216 books.

Check. One-eighth of 216 is 27. Because the part is below 100% of the whole, the recovered whole should exceed 27. The calculation 27 × 0.125 finds a fraction of the part and is not a recovery of the whole.

Worked example 4: a percentage smaller than one

A manufacturing tolerance is 0.6% of a length of 3500 mm. Find the tolerance in millimetres.

Identify. The base is 3500 mm. The decimal rate is 0.6/100 = 0.006.

Calculate. Tolerance = 3500 × 0.006 = 21 mm.

Check. One percent of 3500 mm is 35 mm. Since 0.6% is less than 1%, a tolerance below 35 mm is expected. The unit remains millimetres. Using 0.6 would calculate 60%, not 0.6%.

Worked example 5: name the old base

Daily requests increase from 480 to 540. Find the absolute increase and the percentage increase.

Identify. Old value O = 480; new value N = 540.

Calculate. Absolute increase = 540 − 480 = 60 requests. Percentage increase = 60/480 × 100% = 12.5%.

Check. A 12.5% increase on 480 adds 60, giving 540. Dividing 60 by 540 would measure the gap relative to the new value. It would not answer the question “by what percentage did the original count increase?”

Worked example 6: justify an estimate

Estimate 19.8% of 502, establish a narrow bound, and then find the exact value.

Estimate. Twenty percent of 500 is 100, so a value near 100 is plausible. This benchmark alone does not decide whether the exact value is above or below 100.

Bound. The rate 0.198 is positive and 500 < 502 < 505. Therefore:

0.198 × 500 < 0.198 × 502 < 0.198 × 505

So 99 < answer < 99.99.

Calculate. 0.198 × 502 = (0.2 − 0.002) × 502 = 100.4 − 1.004 = 99.396.

Check. The exact value lies inside the justified interval. Writing 100 as an exact answer would discard information; writing 993.96 would fail the size check immediately.

6. Repair these common errors

Tempting move	Why it fails	Repair
Treat 0.6% as 0.6	The percent sign includes division by 100	Write 0.6/100 = 0.006
Divide the whole by the percentage number to find a part	The model is multiplication by a rate	Write A = B × p/100 first
Multiply a known part by the rate to recover the whole	This takes a smaller part of the part	Divide by the nonzero decimal rate
Divide an increase by the new value	It changes the comparison base	Label old and new before subtracting
Give a count from a rate with no total	Many totals can have the same rate	State which datum is missing
Calculate percentage change from an old value of zero	The required division is undefined	Report absolute change and the limit of the model
Round before a comparison is settled	The rounding error may affect the result	Keep precision or prove a sufficient bound

Before moving on, explain why each denominator in the six cases was chosen. Being able to reproduce an arithmetic step is useful; being able to justify the base lets you handle unfamiliar wording.

7. Fresh practice

Try each question before reading its solution. “Select all” means every correct statement and no incorrect statement. Numeric answers use the unit requested in the prompt.

Question 1

Which percentage is equal to the decimal rate 0.035?

A. 0.35%

B. 3.5%

C. 35%

D. 350%

Question 2

A service completes 18% of its 450 scheduled checks. How many checks does it complete? Enter the number.

Question 3

A device rejects 14 measurements out of 560. What percentage are rejected? Enter the percentage number without the percent sign.

Question 4

171 units form 45% of a shipment. How many units are in the shipment?

Question 5

A stored quantity increases from 320 L to 368 L. What is its percentage increase?

A. Approximately 13.04%

B. 15%

C. 48%

D. 115%

Question 6

A count falls from 750 to 690. What is its percentage decrease? Enter the positive percentage number.

Question 7

Select all correct statements.

A. A compared amount can equal 125% of a positive baseline.

B. 125% of 64 is 80.

C. A fixed group of 64 members can contain 80 distinct members as a subset.

D. 0.8% equals the decimal rate 0.08.

Question 8

Find exactly 12.4% of 802. Before calculating, show that the result lies between 99.2 and 99.82. Enter the exact number.

Question 9

A new counter starts at 0 and later shows 12. Which report is mathematically justified by the usual percentage-change formula?

A. The count increased by 12%.

B. The count increased by 1200%.

C. The absolute increase is 12; percentage increase from this zero base is undefined.

D. The count did not change because the percentage is undefined.

Question 10

Select all conclusions supported by the stated information.

A. If 60% of participants are online, exactly 40 participants must be offline.

B. If 60% of 250 participants are online, 150 are online.

C. If 12 participants are 30% of the total, the total is 40.

D. If zero is 0% of an unknown total, that total must be zero.

8. Solutions and option checks

Solution 1: B

Convert the decimal rate to a percentage: 0.035×100=3.5. Therefore the answer is 3.5%. Reverse check: 3.5/100=0.035.

• A: 0.35%=0.0035, ten times too small.
• B: 3.5%=0.035 exactly.
• C: 35%=0.35, ten times too large.
• D: 350%=3.5, one hundred times too large.

Solution 2: 81

Base=450 checks; rate=0.18. Completed checks=450×0.18=81. For a size check, 20% would be 90, so 81 is plausible.

Solution 3: 2.5

Rate=14/560=1/40=0.025. Multiply by 100 to obtain 2.5%. The numeric entry is 2.5, not 0.025.

Solution 4: 380

Write 171=0.45B. Then B=171/0.45=380. Forward check: 380×0.45=171. The whole is larger than the part.

Solution 5: B

Increase=368−320=48 L. Use the old base: 48/320×100%=15%. The new amount is 115% of the old amount, so its increase is 15%.

• A: 48/368×100 is approximately 13.04%, but it uses the new base.
• B: 48/320×100=15 uses the starting base.
• C: 48 is the increase in litres, not the percentage number.
• D: 115% describes new/old, not the increase/old.

Solution 6: 8

Decrease=750−690=60. Percentage decrease=60/750×100=8%. The signed percentage change would be −8%, but the requested size of the decrease is 8.

Solution 7: A, B

125%=1.25, so 1.25×64=80. A comparison may exceed its baseline, but a subset cannot contain more distinct members than its fixed whole. Also 0.8%=0.008.

• A: Percentages above 100 are valid for comparisons of separate quantities.
• B: 64×1.25=80.
• C: A subset of this fixed 64-member group has at most 64 distinct members.
• D: Dividing 0.8 by 100 gives 0.008, not 0.08.

Solution 8: 99.448

Since 800<802<805 and 0.124>0, multiplying gives 99.2<0.124×802<99.82. Exactly, 0.124×802=0.124×800+0.124×2=99.2+0.248=99.448. It lies within the bound.

Solution 9: C

The absolute increase is 12−0=12. The percentage-change calculation would be 12/0×100%, which is undefined. The lack of a defined ratio does not erase the observed increase.

• A: The absolute increase cannot be relabelled as a percentage without a valid base.
• B: Multiplying by 100 does not resolve division by zero.
• C: It reports the valid absolute change and correctly limits the percentage formula.
• D: The absolute change is known even though the ratio is undefined.

Solution 10: B, C

B follows from 250×0.60=150. C follows from 12/0.30=40. In A, an absolute offline count needs the total and the stated exhaustive two-category model; the percentage alone is insufficient. In D, zero equals 0% of any total, so the total is not determined.

• A: 40% is not the same as 40 people, and no total is supplied.
• B: The known total and rate determine 150.
• C: The known part divided by the nonzero rate determines 40.
• D: For example, both a total of 10 and a total of 100 have a 0% amount of zero.

9. Before you leave

For an unfamiliar percentage problem, say aloud: “My 100% base is …; the amount is …; the rate is …; the unknown is ….” Choose the relationship, estimate, compute and check. If a required denominator is zero or a necessary total is missing, explain the limitation instead of forcing a number.

Next, ratios will express relative shares without immediately turning them into percentages. Keep this lesson's base-first habit: a ratio still needs a scale or a total to determine absolute amounts.

References

Syllabus: GATE 2027 General Aptitude, page 1, section 2.

For further reading: OpenStax Prealgebra 2e, section 6.1, Understand Percent and section 6.2, Solve General Applications of Percent.

Hindi explanations support learning; the GATE 2027 examination is in English.

Analogy

Imagine stretching a strip with 100 equal divisions over the chosen whole. If the whole is a 200 cm ribbon, one division represents 2 cm and 25 divisions represent 50 cm. The same 25% applied to a 40 cm ribbon is 10 cm because the strip is now scaled to a different whole. First choose the ribbon; then count the proportional divisions.

The strip is a model of proportions, not a promise that physical objects can be split into hundredths. A comparison above 100% extends beyond the first strip. The strip cannot supply a missing whole by itself, and a zero-length baseline cannot support the division needed to find a relative percentage.

Quick reference

Name the base first: B is the quantity treated as 100%; A is the compared amount; p is the percentage number.

• Decimal rate = p/100; percentage number = decimal rate × 100
• Amount: A = B × p/100
• Percentage: p = 100A/B, with B ≠ 0
• Whole: B = A ÷ (p/100), with p ≠ 0
• Percentage change = (new − old)/old × 100%, with old ≠ 0
• Percentage decrease = (old − new)/old × 100% when new < old and old > 0
• Keep compared quantities in the same unit. A percentage is dimensionless; an amount keeps the base's unit.
• A comparison may exceed 100%; a fixed whole cannot contain more distinct members than its total.
• Zero old base: report absolute change; the relative percentage change is undefined.
• Zero part at zero rate does not determine the whole. A rate without a total does not determine a count.
• Estimate, preserve precision, and substitute the answer back. Label approximations and round only as requested.

Useful conversions: 1/2 = 50%; 1/4 = 25%; 1/8 = 12.5%; 1/100 = 1%; 0.5% = 0.005. Reference: original teaching based on GATE 2027 GA section 2; concept checks in OpenStax Prealgebra 2e sections 6.1–6.2.