Find your next learning step
Choose the question closest to what you need today, or read the guide from the beginning.
Understand the idea · Compare the examples · See a full solution · Practise and check · Choose the next step
Increase by tells you the amount of change. Increase to tells you the final amount or final level. If forty increases by twelve, the result is fifty-two. If forty increases to fifty-two, the change is twelve. The two phrases can describe the same event, but the number following each phrase plays a different role.
Percentage wording needs one further check: a percentage of what? Increase to twenty per cent may describe a final percentage rate, while increase to twenty per cent of the original describes a fraction of an original quantity. Those statements must not be treated as interchangeable.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–20
Sections 21–25
Sections 26–29
Use the relationship final amount = original amount + change. The word by usually supplies the change in an increase instruction. The word to supplies the endpoint. For an original amount of forty, increased by fifty-two gives ninety-two, while increased to fifty-two gives fifty-two.
With percentages, an increase by twenty per cent of the original means adding 0.2 times the original. The final amount is 1.2 times the original. An amount changed to one hundred and twenty per cent of the original also becomes 1.2 times the original.
Do not ignore the reference phrase. A rate increasing from ten per cent to twenty per cent ends at twenty per cent; it does not become twenty per cent of ten per cent. The wording identifies whether the percentage is itself the quantity or is describing a fraction of another quantity.
This is a reading distinction with mathematical consequences. It appears in arithmetic questions about quantities, percentage-change problems, algebraic models, graph descriptions and geometric scale factors. Correct calculation begins with deciding which number is an original amount, a change or a final amount.
The SEAB 2027 G1 Mathematics syllabus includes percentages and practical numerical work. The same reading discipline remains useful as questions become more algebraic at other subject levels. The examples here support understanding rather than prescribe one identical syllabus sequence for every learner.
For broader language support, connect this article to eduKate’s explanation of mathematics command words and how they change the operation. That guide covers a wider range of instructions. Here we focus closely on by, to and the reference quantity in percentage statements.
All numerical situations in this guide are invented teaching examples. They are not offers, quoted prices, financial advice or claims about actual student outcomes. Their purpose is to make the roles of the numbers easy to compare.
Contents · Previous section · Next section
Write O for the original amount, C for the signed change and F for the final amount. Their relationship is F = O + C. For an increase, C is positive. For a decrease, the signed change is negative, even if a question describes the size of the decrease as a positive number.
If O and C are known, add to find F. If O and F are known, subtract to find C = F − O. If F and C are known, subtract the change to recover O = F − C. These are rearrangements of one relationship, not three unrelated tricks.
The phrase increased by twelve normally identifies C = 12. The phrase increased to fifty-two identifies F = 52. The number twelve and the number fifty-two are therefore not automatically placed into the same position in an equation.
Before calculating, write the labels above or beside the numbers. A few seconds spent identifying their roles is often more useful than immediately deciding whether a familiar word must mean addition. The instruction’s full relationship determines the operation needed to answer the particular question.
Contents · Previous section · Next section
An invented display has forty tiles. Its tile count increases by twelve. The change is twelve tiles, so the final count is 40 + 12 = 52 tiles. The original forty tiles remain included in the final fifty-two.
If the question instead says the count increases by fifty-two, the final count is 40 + 52 = 92. Using fifty-two as the endpoint would ignore the meaning of by and discard part of the requested increase.
Units help keep the statement precise. Twelve tiles is an absolute change. Twelve per cent is a relative description that needs a reference quantity. Both can follow by, but they do not supply the same kind of numerical input.
To check an absolute increase, subtract the original from your proposed final amount. If 52 − 40 = 12, the final amount matches an increase by twelve. If the difference does not equal the stated increase, revisit the role you assigned to that number.
Contents · Previous section · Next section
Suppose the tile count increases from forty to fifty-two. The endpoint is already known: F = 52. If the question asks how many tiles were added, calculate 52 − 40 = 12. If it asks the final count, answer fifty-two without adding the original again.
The phrase increased to fifty-two does not mean that fifty-two tiles were added. It means the count reached fifty-two. Treating an endpoint as an increment would produce ninety-two, which is a different final amount and a different event.
A number line can support the interpretation: forty is the start, fifty-two is the finish and twelve is the distance between them. The word to points towards the finish rather than describing the length of the movement.
This structure also clarifies incomplete information. If you know only that an amount increased to fifty-two, you know its final amount but not its original amount or exact increase. An endpoint alone does not reveal the journey’s starting point.
Contents · Previous section · Next section
| Instruction with original amount 40 | Change | Final amount |
|---|---|---|
| Increase by 12 units | 12 units | 52 units |
| Increase to 52 units | 12 units | 52 units |
| Increase by 52 units | 52 units | 92 units |
| Increase to 60 units | 20 units | 60 units |
| Increase by 60 units | 60 units | 100 units |
The first two rows describe the same numerical event using different supplied information. The third row shows why copying the number after to into a by calculation changes the answer. The label attached to a number matters as much as the number itself.
When comparing two instructions, do not look only for repeated digits. Translate each instruction into O, C and F. Then decide whether they describe the same final amount, the same change, or neither. This is a transferable reading method rather than a one-example memory aid.
Contents · Previous section · Next section
Suppose a ribbon originally measures eighty centimetres and its length increases by twenty per cent of that original length. Twenty per cent of eighty is 0.2 × 80 = 16 centimetres. The final length is 80 + 16 = 96 centimetres.
The original amount represents one hundred per cent of itself. Adding another twenty per cent gives a final amount equal to one hundred and twenty per cent of the original. This leads to the multiplier 1.2 and the compact calculation 1.2 × 80 = 96.
The increase is sixteen centimetres, not twenty centimetres. A percentage describes a proportion of its reference quantity. Unless the original happens to be one hundred units, the percentage number and the absolute increase will not have the same numerical value.
Use a two-step calculation if the multiplier is not yet meaningful: first find the change, then add it. The multiplier is a shorter expression of the same reasoning, not a replacement for understanding where the extra part comes from.
Contents · Previous section · Next section
If the ribbon becomes one hundred and twenty per cent of its original eighty-centimetre length, the final length is 1.2 × 80 = 96 centimetres. That matches an increase by twenty per cent of the original.
If it becomes one hundred and fifty per cent of the original, the final multiplier is 1.5. The increase is fifty per cent of the original, because 150% − 100% = 50%. It is not an increase by one hundred and fifty per cent.
For a positive original quantity, becoming twenty per cent of the original gives a smaller final amount, not an increase. From eighty centimetres, that final amount is sixteen centimetres. A statement calling this an increase would need clarification or would be inconsistent with that interpretation.
The phrase of the original does crucial work. Without it, to twenty per cent might instead name a new rate, share or test reading. Do not insert an unstated reference quantity simply because a familiar multiplier pattern is available.
Contents · Previous section · Next section
Suppose an invented process has a completion rate that rises from ten per cent to twenty per cent. The original quantity being discussed is itself a percentage rate. Its final level is twenty per cent, not two per cent.
The increase is ten percentage points, because 20% − 10% = 10 percentage points. Relative to the original ten per cent, that increase is one hundred per cent: the rate has doubled. These are two different descriptions of the same change.
By contrast, twenty per cent of an original ten-per-cent rate is two per cent. That is a fraction of the original rate, not the stated endpoint twenty per cent. Confusing the two readings changes both the size and direction of the event.
When an isolated phrase says increased to twenty per cent, ask what is being measured. If it names a rate’s final value, use twenty per cent as that value. If it explicitly says twenty per cent of an original amount, apply that stated reference. Missing context cannot be repaired by a universal keyword rule.
Contents · Previous section · Next section
An increase by one hundred and fifty per cent adds one and a half times the original amount to the original itself. For an original amount of forty units, the increase is 1.5 × 40 = 60 units. The final amount is 100 units.
The final multiplier is 1 + 1.5 = 2.5. This is different from increasing to one hundred and fifty per cent of the original, which uses multiplier 1.5 and gives sixty units from the same original forty.
There is no rule that an increase percentage must be below one hundred. An increase above one hundred per cent simply adds more than the original amount. The calculation still follows original plus the stated proportion of original.
When the percentage is large, write the one hundred per cent original part explicitly. It makes the difference between final share and added share visible. The number after by describes the addition; the total includes the original as well.
Contents · Previous section · Next section
Let the original positive amount be O and let r be a percentage expressed as a decimal. An increase by r of the original gives F = O + rO = (1 + r)O. If the increase is twenty per cent, use r = 0.2.
If the final amount is k times the original, then F = kO. The relative change is (F − O)/O = k − 1, provided O is not zero. Thus a final multiplier of 1.2 corresponds to relative increase 0.2, or twenty per cent.
If a statement explicitly gives the final amount as p per cent of the original, its multiplier is p/100. The increase percentage is p − 100 percentage units of the original percentage scale; in ordinary wording, it is an increase of (p − 100)% when p exceeds one hundred.
For a positive original, a multiplier below one gives a decrease, one gives no change and above one gives an increase. This check can reveal a contradiction between a calculated multiplier and the instruction’s claimed direction before further work is done.
Contents · Previous section · Next section
Suppose the final ribbon length is ninety-six centimetres after an increase by twenty per cent of the original. The final amount represents one hundred and twenty per cent of the original, so 1.2O = 96. Divide to obtain O = 96/1.2 = 80 centimetres.
Subtracting twenty per cent of ninety-six would give 76.8 centimetres, which is not the original. That subtraction uses the final amount as its percentage base, while the increase was defined using the original amount. The reference quantity changed without permission.
Check the recovered original by applying the forward instruction: twenty per cent of eighty is sixteen, and eighty plus sixteen is ninety-six. A reverse answer should survive the original forward relationship.
If the final amount is explicitly one hundred and twenty per cent of the original, the same equation 1.2O = 96 applies. Different wording can describe an identical relationship, provided the endpoint percentage and the increase percentage are matched correctly.
Contents · Previous section · Next section
An amount increases from forty units to sixty units. Its absolute increase is 60 − 40 = 20 units. To express that increase as a percentage of the original, divide by forty and multiply by one hundred: 20/40 × 100% = 50%.
The final amount is sixty fortieths of the original, or one hundred and fifty per cent. Therefore increased by fifty per cent and increased to one hundred and fifty per cent of the original describe this same event.
Using sixty as the denominator instead would answer a different question: the increase is one third of the final amount. That is not the usual original-based percentage increase. Name the reference quantity before choosing the denominator.
The unit count and the percentage change are different answers. Twenty units and fifty per cent both describe the same increase, but they are not interchangeable labels. Read whether the question asks for an amount, a percentage or a final value.
Contents · Previous section · Next section
Question: a design begins with a length of fifty centimetres. Version A increases the length by twenty per cent of its original. Version B increases it to one hundred and twenty centimetres. Find each final length and each absolute increase.
For Version A, the change is 0.2 × 50 = 10 centimetres. The final length is sixty centimetres. For Version B, the final length is already stated as one hundred and twenty centimetres. Its increase is 120 − 50 = 70 centimetres.
If we also express Version B’s increase relative to the original, we obtain 70/50 × 100% = 140%. Its final length is two hundred and forty per cent of the original, because 120/50 = 2.4.
These answers show why the words and units must be read together. Twenty per cent, one hundred and twenty centimetres, one hundred and forty per cent increase and two hundred and forty per cent of original all identify different mathematical roles. None should be substituted for another because the digits look familiar.
Contents · Previous section · Next section
For each question, write the original, change and final amount where they are known. State the percentage base when a percentage is used. Then calculate the requested quantity. The aim is a correct relationship before arithmetic speed.
Question A. Thirty-five units increase by nine units. Find the final amount. Question B. Thirty-five units increase to forty-four units. Find the increase. Question C. Eighty units increase by twenty-five per cent of the original. Find the increase and final amount.
Question D. Eighty units become one hundred and twenty-five per cent of the original. Find the final amount. Question E. Forty units increase by one hundred and fifty per cent of the original. Compare this with becoming one hundred and fifty per cent of the original.
Question F. A final amount is seventy-two after an increase by twenty per cent of the original. Find the original amount. Question G. A rate increases from fifteen per cent to twenty per cent. Find the increase in percentage points and its relative percentage increase.
Question H. An amount increases from fifty to sixty-five. Find its absolute and percentage increase. Question I. A positive amount decreases to twenty-five per cent of the original. State the decrease percentage. Question J. Explain why increased to twenty per cent needs context before it can be translated into a multiplier of an original quantity.
Contents · Previous section · Next section
Answer A. The final amount is 35 + 9 = 44 units. Answer B. The increase is 44 − 35 = 9 units. The same numerical event is described by a change in A and an endpoint in B.
Answer C. Twenty-five per cent of eighty is twenty, so the final amount is one hundred. Answer D. The final amount is 1.25 × 80 = 100. Becoming one hundred and twenty-five per cent of the original matches an increase by twenty-five per cent.
Answer E. Increasing by one hundred and fifty per cent adds sixty to forty, giving one hundred. Becoming one hundred and fifty per cent of the original gives sixty, an increase of twenty. The respective multipliers are 2.5 and 1.5.
Answer F. The original is 72/1.2 = 60. Checking forward, twenty per cent of sixty is twelve and sixty plus twelve is seventy-two. Answer G. The rate rises by five percentage points. Relative to the original fifteen per cent, the increase is 5/15 = 1/3, or 33⅓%.
Answer H. The absolute increase is fifteen. Relative to the original fifty, it is 15/50 × 100% = 30%. Answer I. The final amount is one quarter of the original, so three quarters has been removed: the decrease is seventy-five per cent of the original.
Answer J. Twenty per cent might name a final rate or might be part of a phrase such as twenty per cent of the original. A final rate of twenty per cent is not automatically twenty per cent multiplied by a previous rate. The measured quantity and reference phrase determine the reading.
Contents · Previous section · Next section
For an original positive amount of eighty units, decreasing by twenty units gives sixty. Decreasing to twenty units gives twenty, with a decrease of sixty. The prepositions still separate a change from an endpoint.
Decreasing by twenty-five per cent of the original leaves seventy-five per cent of the original, so the final amount is 0.75 × 80 = 60. Decreasing to twenty-five per cent of the original leaves only twenty units, a decrease of sixty.
The final multipliers are therefore 0.75 and 0.25. A common error is to use the percentage following to as the removed share. That reverses remaining and removed, which are complementary parts of the original whole.
Check by adding the removed amount to the remaining amount. They should reconstruct the original. If eighty becomes twenty, sixty was removed. If eighty loses twenty, sixty remains. A reconstruction check keeps those two statements distinct even when the same numbers appear.
Contents · Previous section · Next section
Start with one hundred units and increase it by twenty per cent of the current amount. It becomes one hundred and twenty. Increase this new amount by twenty per cent of its current value, and the second increase is twenty-four, giving one hundred and forty-four.
The combined multiplier is 1.2 × 1.2 = 1.44. The total increase is forty-four per cent of the original, not forty per cent. The second percentage was applied to a larger base than the first.
If a question instead states that two separate additions are each twenty per cent of the same original one hundred, the final amount would be one hundred and forty. That is a different instruction with a fixed shared reference quantity.
Words such as current, original, then and again deserve attention. They tell you whether the percentage base updates after each step. Write a short sequence of amounts when necessary, so the reference quantity for each change remains explicit.
Contents · Previous section · Next section
Begin with eighty units. Increasing by twenty-five per cent of the current amount gives one hundred. Decreasing that new amount by twenty-five per cent gives seventy-five. The final amount is five below the original, not back at eighty.
The combined multiplier is 1.25 × 0.75 = 0.9375. Equal percentage numbers applied to different bases do not imply equal absolute changes. The initial increase was twenty units, while the later decrease was twenty-five units.
To return from one hundred to eighty, the required decrease is twenty per cent of one hundred. Reversing the multiplier means dividing by 1.25, or multiplying by 0.8, rather than subtracting the same stated percentage indiscriminately.
This example reinforces the main reading habit. Identify the current base at each step before interpreting by as a numerical change. The operation depends on both the percentage and the amount to which that percentage refers.
Contents · Previous section · Next section
Suppose orange beads form twenty per cent of a collection, and later they form thirty per cent. The share has increased by ten percentage points. Its relative increase is fifty per cent because ten is half the original twenty.
These statements alone do not determine how many orange beads were added. The total collection might have changed, other beads might have been removed, or both events might have occurred. A change in share does not uniquely specify the underlying count changes.
If the total is fixed at one hundred beads, the counts would move from twenty to thirty, an increase of ten. If the totals differ, additional information is needed. State that limitation instead of inventing a constant whole because the percentages are convenient.
This is where language and ratios meet. A percentage share contains a numerator and a reference whole. Changing the whole can alter the share without the numerator moving in the assumed way. The mathematics learning hub connects the ratio and percentage guides for this reason.
Contents · Previous section · Next section
An invented square has side length ten centimetres. Its side length increases to one hundred and twenty per cent of the original, so the new side is twelve centimetres. The side multiplier is 1.2 and the side increase is twenty per cent.
The area changes from one hundred square centimetres to one hundred and forty-four square centimetres. The area multiplier is 1.2² = 1.44. Thus area increases by forty-four per cent, not twenty per cent.
The wording must identify whether length, area or volume is changing. A scale factor for one quantity cannot simply be copied into an answer about another. The quantity’s mathematical relationship determines how its change is calculated.
If a problem says the area increases by twenty per cent, that supplies an area multiplier of 1.2. It does not supply a side multiplier of 1.2. For similar squares the side multiplier would be √1.2. Keep the measured quantity attached to every percentage statement.
Contents · Previous section · Next section
A graph might show a quantity rising from forty to sixty. The vertical coordinates identify the original and final levels. Their difference, twenty, is the change. The endpoint sixty is not itself the increase unless the original level was zero.
If the horizontal interval is four time units, the average rate of change is twenty divided by four, or five quantity-units per time-unit. This introduces another role: the change per unit of the horizontal variable. It is neither the final level nor the total increase.
A carefully labelled answer separates all three: final level sixty units, increase twenty units, average rate five units per time-unit. The units make it harder to substitute one number for another without noticing.
When a graph’s vertical axis shows percentages, use the same discipline. A rise from ten per cent to twenty per cent is a ten-percentage-point change in the vertical level. Its relative increase and its rate per horizontal unit are separate calculations.
Contents · Previous section · Next section
The ordinary relative percentage-change formula divides by the original amount. If the original is zero, that division is undefined. A quantity moving from zero to ten has an absolute increase of ten, but the usual original-based percentage increase is not a finite number obtained by the standard formula.
For negative original quantities, casual percentage-growth wording can be misleading. Multiplying a negative number by 1.2 makes it more negative, which is numerically a decrease. A problem must define the quantity and intended interpretation clearly rather than rely on the usual positive-amount intuition.
This article’s ordinary increase-percentage examples therefore use positive originals. That condition is not an irrelevant technicality; it makes the statements about multipliers above and below one align with numerical increase and decrease.
If a question involves signed balances, directed quantities or a zero starting value, write the actual original and final values first. Calculate absolute change directly, and only use a relative percentage definition when its denominator and interpretation are valid and stated.
Contents · Previous section · Next section
The statement forty units increased to ten units is inconsistent if it refers to the same ordinary quantity and no other interpretation is supplied. Ten is less than forty. The arithmetic does not become an increase merely because the word increased appears in the sentence.
Likewise, a positive amount described as increasing to twenty per cent of its original would actually become smaller under that explicit reference. Flag the inconsistency and ask whether the intended wording was increased by twenty per cent or decreased to twenty per cent.
Do not silently replace the instruction and present your replacement as the only possible answer. If a worksheet contains a probable typo, state the assumption under which you proceed. That preserves the distinction between given information and a reasonable repair.
This is part of mathematical reading, not a refusal to calculate. Sometimes the correct conclusion is that the information is ambiguous or inconsistent. A clear explanation of the missing reference or conflicting direction is more honest than a confident number based on an unstated guess.
Contents · Previous section · Next section
Suppose a student calculates eighty increased by twenty per cent as sixteen. The percentage calculation is correct, but sixteen is the increase rather than the final amount. The repair is to add the original eighty, not to reteach the multiplication as though it were wrong.
If another student calculates eighty increased to one hundred as one hundred and eighty, the problem is different. The student has treated a final endpoint as an added change. Write F = 100 first, then find C = 100 − 80 if the change is requested.
If a student reverses a twenty-per-cent increase by subtracting twenty per cent of the final amount, identify the reference-base error. A multiplier equation makes the intended relationship visible and allows the reverse operation to be chosen correctly.
Correcting the first wrong relationship is more precise than giving a general instruction to read carefully. Name what the number represents, show where it belongs in the equation, and then check whether the arithmetic following that equation is consistent.
Contents · Previous section · Next section
Read the full statement once without calculating. On the second pass, mark from or originally as the start, by as a stated change, and to as a stated endpoint. Treat these as prompts to interpret the sentence, not automatic rules that override its context.
Circle units and reference phrases, especially of the original, of the current amount and percentage points. If a percentage has no clear base, identify whether it is itself the measured level or a proportion applied to another quantity.
Write one relationship before using a calculator. For example, final = 1.2 × original expresses a twenty-per-cent increase of a positive original. Final = 120 units expresses a stated endpoint. Those two lines cannot be substituted for each other.
Check the direction and size of the result. For a positive original, an increase should leave a larger final amount. Reverse the calculation when possible, and label whether your answer is a change, endpoint, percentage or rate. Those small checks make the explanation readable to another person as well as correct.
Contents · Previous section · Next section
If an unknown positive amount increases by twenty per cent, its final multiplier is known: 1.2. The absolute increase and final number of units are not known without an original amount or equivalent information. A relative answer can be determined while an absolute answer remains undetermined.
For example, an original of fifty gives a change of ten and final amount sixty. An original of one hundred gives a change of twenty and final amount one hundred and twenty. Both satisfy the same twenty-per-cent increase, so neither numerical final amount can be inferred from the percentage alone.
If an amount increases to sixty units, the final amount is known even if the starting amount is missing. The increase might be ten units from fifty, or twenty units from forty. The endpoint does not determine the change without a start.
Before writing insufficient information, identify what the question requests. A missing original blocks some answers but not every answer. State which quantity remains free and, if useful, show two valid starting amounts that produce different possible changes or endpoints.
Contents · Previous section · Next section
Begin with O = 40, C = 12 and F = 52. One accurate sentence is forty increases by twelve to fifty-two. Another is the final amount exceeds the original by twelve. Both preserve the same relationship while assigning different grammatical positions to the numbers.
Now express the relative change. Twelve is thirty per cent of forty, so the amount increases by thirty per cent of the original. Fifty-two is one hundred and thirty per cent of forty, so the final amount is one hundred and thirty per cent of the original.
Translating an equation back into a sentence is a strong check on interpretation. If your sentence says increased by fifty-two while your equation uses fifty-two as F, the wording and algebra disagree. Correct one before continuing the arithmetic.
For revision, ask someone to change only by to to while keeping the following number fixed. Explain what information has changed, then recalculate where possible. This small controlled change tests whether the preposition’s role is understood instead of recognised only in a memorised example.
Contents · Previous section · Next section
These final checks emphasise meaning before arithmetic. Identify whether the number is a change or endpoint, and whether a percentage names the quantity itself or a proportion of another clearly specified reference quantity.
Does by always mean I should add the following number?
It usually identifies the change in an increase statement, but a percentage must first be applied to its reference quantity. Increased by twenty per cent does not mean add twenty units. Read the whole instruction and its units before selecting an operation.
Is increased to one hundred and twenty per cent the same as increased by twenty per cent?
Yes when the first phrase explicitly means one hundred and twenty per cent of the same original positive amount. Without the reference, one hundred and twenty per cent could be a final rate or level in a different comparison. State what the percentage describes.
Why can increased to twenty per cent be ambiguous?
It may name a final percentage rate, such as a share rising from ten per cent to twenty per cent. It does not automatically mean twenty per cent of the previous rate or of an original amount. The missing noun and reference phrase matter.
What should I remember first?
By identifies a change; to identifies an endpoint. Percentages require a reference quantity. Put original, change and final into one labelled relationship, then calculate the quantity the question actually requests. That habit works across arithmetic, algebra, graphs and measurement.
Continue the G1/G2/G3 Mathematics series
Browse every guide in the G1/G2/G3 Mathematics directory
What Is G1 Mathematics | Why a Ratio of 2:3 Is Not Two Thirds of the Combined Total
What Is G2 Mathematics | Why a Negative Gradient Does Not Mean Every y-Value Is Negative
What Is G3 Mathematics | Why Adding Five to Every Value Does Not Add Five to Standard Deviation