Percentage Points

When the quantity that changes is itself a percentage, such as a pass rate or the GST rate: its change in percentage points, its percentage change, and how to tell which one a statement means.

Before you start

A percentage change compares a change with the original value: percentage change $= \frac{\text{change}}{\text{original value}} \times 100\%$ .

For example, a plate of chicken rice that cost $4 now costs $5. The change is $1, so the percentage increase is $\frac{1}{4} \times 100\% = 25\%$ of the original price.

An answer that is not exact is given correct to 3 significant figures, with "≈" before it.

Two Ways to Describe a Change in a Percentage

Many numbers in everyday life are percentages already: the pass rate of a test, the GST rate, the interest rate on a savings account, the share of the votes that a candidate wins, the percentage of students who walk to school. Such a percentage can change like any other quantity. On 1 January 2024 the GST rate in Singapore went up from 8% to 9%. By how much did it go up?

There are two true answers, and they measure two different things. One is how far apart the two percentages are. The other is how big the change is compared with the original percentage. The first needs a unit of its own, the percentage point.

Definition: percentage points and percentage change

When a percentage changes, the difference between the new percentage and the original percentage is measured in percentage points:

$\text{change in percentage points} = \text{new percentage} - \text{original percentage}$

If the answer is negative, the percentage has fallen: from 40% to 30% is $30 - 40 = -10$ , a fall of 10 percentage points. Use the size of the fall, 10, for the percentage decrease, as for any percentage decrease.

The percentage change compares that difference with the original percentage, as every percentage change does:

$\text{percentage increase or decrease} = \frac{\text{rise or fall in percentage points}}{\text{original percentage}} \times 100\%$

For example, a percentage that goes from 20% to 25% has risen by $25 - 20 = 5$ percentage points, and it has risen by $\frac{5}{20} \times 100\% = 25\%$ .

When a percentage such as a rate or a share changes from an old value to a new one, the change is stated in percentage points, not with "%". If both percentages are of the same base, 1 percentage point is 1% of that base.

Example 1: The GST rate in 2024

On 1 January 2024 the GST rate in Singapore went up from 8% to 9%.
(a) By how many percentage points did the GST rate go up?
(b) By what percentage did the GST rate go up?

Solution

(a) The change in a rate is measured in percentage points. Subtract the original rate from the new rate: $9 - 8 = 1$ .

(b) The percentage change compares the change, 1 percentage point, with the original rate, 8%.

Percentage increase $= \frac{1}{8} \times 100\% = 12.5\%$

(a) The GST rate went up by 1 percentage point.
(b) The GST rate went up by 12.5%: the new rate is 12.5% more than the old rate.

A bar model shows both answers at once. Draw the old rate as 8 equal units and the new rate as 9 of the same units, so that 1 unit stands for 1 percentage point.

The 2024 bar has 1 more unit (shaded). Counted in units, the change is 1 percentage point. Compared with the 8 units of the old rate, the extra unit is $\frac{1}{8}$ of it, and $\frac{1}{8} = 12.5\%$ . So "up 1 percentage point" and "up 12.5%" describe the same change. They answer two different questions: "How far apart are the two rates?" and "How big is the change compared with the old rate?"

What does this mean for a shopper? For a $200 purchase, 1 percentage point of GST is 1% of $200, which is $2. The GST on it went up from $16 to $18, and $2 is 12.5% of the old GST of $16.

Percentage Points and Percentage Change Side by Side

The table puts the two measures side by side for six changes. The percentage points come from subtracting; the percentage change comes from dividing that difference by the original percentage.

From	To	Change in percentage points	Percentage change
8%	9%	up 1	$\frac{1}{8} \times 100\% = 12.5\%$ increase
7%	8%	up 1	$\frac{1}{7} \times 100\% \approx 14.3\%$ increase
20%	25%	up 5	$\frac{5}{20} \times 100\% = 25\%$ increase
50%	55%	up 5	$\frac{5}{50} \times 100\% = 10\%$ increase
2%	4%	up 2	$\frac{2}{2} \times 100\% = 100\%$ increase
40%	30%	down 10	$\frac{10}{40} \times 100\% = 25\%$ decrease

Two things stand out. First, the same number of percentage points can be very different percentage changes: a rise of 5 percentage points is a 25% increase from 20%, but only a 10% increase from 50%.

Second, the smaller the original percentage, the bigger the percentage change for the same number of points. From 2% to 4% is a rise of only 2 percentage points, yet the percentage has doubled, which is a 100% increase.

Example 2: A pass rate

Last year 60% of the Secondary 1 students of a school passed a swimming test. This year 75% passed.
(a) By how many percentage points did the pass rate rise?
(b) Find the percentage increase in the pass rate.
(c) There were 200 Secondary 1 students in each year. How many more students passed this year? Express this as a percentage of the number who passed last year.

Solution

(a) Change in percentage points $= 75 - 60 = 15$

(b) The original pass rate is 60%. Percentage increase $= \frac{15}{60} \times 100\% = 25\%$

The bar model shows why. Let 1 unit stand for 5 percentage points. Last year's rate, 60%, is 12 units and this year's rate, 75%, is 15 units. The 3 extra units are the 15 percentage points, and 3 units out of 12 is $\frac{3}{12} = \frac{1}{4}$ of last year's rate, which is 25%.

(c) Last year: $\frac{60}{100} \times 200 = 120$ students passed. This year: $\frac{75}{100} \times 200 = 150$ students passed.

$150 - 120 = 30$ , so 30 more students passed.

As a percentage of last year's number: $\frac{30}{120} \times 100\% = 25\%$

Check: 1 percentage point of 200 students is 2 students, so 15 percentage points is $15 \times 2 = 30$ students.

(a) The pass rate rose by 15 percentage points.
(b) The pass rate rose by 25%.
(c) 30 more students passed this year, which is 25% more than last year.

Example 3: A fall

In a survey of a school's students, 36% walked to school in 2024 and 27% walked to school in 2025. Describe the change in two ways.

Solution

Change in percentage points $= 27 - 36 = -9$ . The minus sign shows a fall: a fall of 9 percentage points.

The original percentage is 36%. Percentage decrease $= \frac{9}{36} \times 100\% = 25\%$

The percentage of students who walk to school fell by 9 percentage points. It also fell by 25%: the new percentage is 25% less than the 36% it started from.

Example 4: A share of a budget

A school's CCA budget is $ $50\,000$ a year. One year it spends 6% of the budget on sports equipment. The next year the budget is the same, and it spends 8.5% of it on sports equipment.
(a) By how many percentage points did the share spent on sports equipment go up?
(b) Find the amount spent on sports equipment in each year, and the percentage increase in that amount.
(c) Find the percentage increase in the share, and compare it with your answer to (b).

Solution

(a) Change in percentage points $= 8.5 - 6 = 2.5$ . The share went up by 2.5 percentage points, not "by 2.5%".

(b) The first year: $\frac{6}{100} \times 50\,000 = 3000$ , so $3000.

The next year: $\frac{8.5}{100} \times 50\,000 = 4250$ , so $4250.

Increase in the amount: $4250 - 3000 = 1250$ , so $1250.

Percentage increase in the amount $= \frac{1250}{3000} \times 100\% = 41.666\ldots\% \approx 41.7\%$

The two percentage increases are the same. The budget did not change, so each percentage point of the share is always the same sum of money: 1% of $ $50\,000$ , which is $500. The rise of 2.5 percentage points is $2.5 \times 500 = 1250$ , that is, $1250: the increase found in (b).

(a) The share went up by 2.5 percentage points.
(b) The amount spent on sports equipment went up from $3000 to $4250, an increase of about 41.7%.
(c) The share went up by about 41.7%, the same as the amount spent, because the budget stayed the same.

When the base stays the same

In Examples 2 and 4 the base did not change: 200 students in each year, and a budget of $ $50\,000$ in each year. Then each percentage point is always the same amount (2 students; $500), and the percentage change of the rate or the share is also the percentage change of the amount it gives (the number of passes; the amount spent).

When the base changes, the two percentage changes are no longer the same. If more students take the test this year, the number of passes rises faster than the pass rate; if fewer take it, the number of passes rises more slowly than the pass rate, and may even fall.

Reading Statements: Which Is Meant?

Newspapers, school reports and advertisements describe changes in percentages every day. To check a statement, find the two percentages, work out both the change in percentage points and the percentage change, and see which one the statement gives.

How to read a statement

"Rose by 3 percentage points" (sometimes shortened to "3 points") means that the difference of the two percentages is 3: subtract.

"Rose by 3%" means, in mathematics, that the new percentage is 3% more than the original percentage: compare the change with the original by dividing.

Many writers use "%" when they mean percentage points. When a statement uses "%" for the change of a percentage, check it with the numbers before you trust it.

In this lesson's questions, a statement is judged by its mathematical meaning: "up 3%" means up 3% of the old percentage. Example 7 shows the other reading that an everyday reader may take.

Example 5: The GST rate in 2023

On 1 January 2023 the GST rate in Singapore went up from 7% to 8%. Say whether each statement is right or wrong, and correct the one that is wrong.
(a) "The GST rate went up by 1 percentage point."
(b) "The GST rate went up by 1%."
(c) "The GST rate went up by about 14.3%."
(d) "From 2022 to 2024 the GST rate went up by 2 percentage points, an increase of about 28.6%."

Solution

Find the two measures first. Change in percentage points: $8 - 7 = 1$ . Percentage increase: $\frac{1}{7} \times 100\% = 14.285\ldots\% \approx 14.3\%$

(a) Right: the difference of the two rates is 1 percentage point.

(b) Wrong. In mathematics "went up by 1%" means up by 1% of 7%, which is $\frac{1}{100} \times 7 = 0.07$ of a percentage point, to 7.07%. The rate went up by 1 percentage point, which is about 14.3% of the old rate.

(d) The rate was 7% in 2022 and 9% in 2024. Change in percentage points: $9 - 7 = 2$ . Percentage increase: $\frac{2}{7} \times 100\% = 28.571\ldots\% \approx 28.6\%$ . Right.

(a), (c) and (d) are right. (b) is wrong: it should say "The GST rate went up by 1 percentage point" or "The GST rate went up by about 14.3%".

Example 6: A survey

In a school survey, 35% of the students named basketball as their favourite sport last year. This year 42% did. The school newsletter says: "Basketball's share went up by 7%." Is this right? Write two correct statements about the change.

Solution

Change in percentage points: $42 - 35 = 7$

Percentage increase: $\frac{7}{35} \times 100\% = 20\%$

The newsletter wrote the change in a percentage with "%". The change in a percentage is in percentage points. Read in the mathematical way, "up by 7%" would mean up by 7% of 35%, which is $\frac{7}{100} \times 35 = 2.45$ percentage points, to 37.45%. That is not what happened.

No. Two correct statements: "Basketball's share went up by 7 percentage points, from 35% to 42%" and "Basketball's share went up by 20%".

Making an Unclear Statement Clear

When the thing that changes is itself a percentage, a sentence such as "the pass rate rose by 5%" can be read in two ways. Some readers add 5 percentage points; others work out 5% of the old rate. A clear statement leaves only one reading.

Example 7: Two readings

(a) A school's pass rate in a test was 50% last year. A report says that this year "the pass rate rose by 5%". What are the two pass rates this could mean? Rewrite the sentence clearly for each.
(b) Last term 30% of the students of a school bought a drink at the canteen each day. A report says that this term "the percentage fell by 10%". What are the two percentages this could mean?

Solution

(a) Reading 1, "rose by 5 percentage points": $50 + 5 = 55$ , so the pass rate is 55%.

Reading 2, "rose by 5% of the old rate": 5% of 50% is $\frac{5}{100} \times 50 = 2.5$ percentage points, so the pass rate is $50 + 2.5 = 52.5$ , that is, 52.5%.

Clear sentences say which: "The pass rate rose by 5 percentage points, from 50% to 55%" or "The pass rate rose by 5%, from 50% to 52.5%".

(b) Reading 1, "fell by 10 percentage points": $30 - 10 = 20$ , so 20%.

Reading 2, "fell by 10% of the old percentage": 10% of 30% is $\frac{10}{100} \times 30 = 3$ percentage points, so $30 - 3 = 27$ , that is, 27%.

(a) 55% or 52.5%: "The pass rate rose by 5 percentage points, from 50% to 55%" or "The pass rate rose by 5%, from 50% to 52.5%".
(b) 20% or 27%.

Writing the change of a percentage clearly

Say "percentage points" for the change in a percentage, and keep "%" for a percentage change.

Better still, give both percentages: "from 50% to 55%". Then a reader can work out either measure.

Why a Writer Might Choose One or the Other

Both measures are true, but they can sound very different. A writer who wants a change to sound big tends to choose the bigger number, and one who wants it to sound small chooses the smaller. When the original percentage is small, the percentage change is much bigger than the change in points.

Example 8: Big or small?

(a) At a school, 4% of the students were late at least once in Term 1, and 2% in Term 2. Describe the fall in two ways. Which would the principal choose when she praises the students, and why?
(b) Last year 1% of the students of a school cycled to school; this year 3% did. A cycling club writes "Cycling to school up 200%!" A student writes "Cycling to school up only 2 points." Are both right? Write a fair headline.

Solution

(a) Change in percentage points: $2 - 4 = -2$ , a fall of 2 percentage points.

Percentage decrease: $\frac{2}{4} \times 100\% = 50\%$ . The percentage of students who were late has halved.

"Lateness fell by 50%" (or "halved") sounds far better than "lateness fell by 2 percentage points", and both are true.

(b) Change in percentage points: $3 - 1 = 2$ . Percentage increase: $\frac{2}{1} \times 100\% = 200\%$ . The percentage has tripled.

Both are right, though "2 points" should say "2 percentage points". The club chooses the big number and the student chooses the small one.

A fair headline gives the two percentages, so that a reader sees both that the share has tripled and that it is still small.

(a) A fall of 2 percentage points, or a fall of 50%. The principal would choose "fell by 50%" (or "halved") because it sounds like a bigger improvement.
(b) Both are right. A fair headline: "Cycling to school up from 1% to 3% of students".

Common Mistakes

Writing "%" for the change in a percentage. A rate that goes from 6% to 8.5% has not "risen by 2.5%". It has risen by 2.5 percentage points, which is an increase of about 41.7% on the old rate.
Dividing by the new percentage. From 8% to 9% is an increase of $\frac{1}{8} \times 100\% = 12.5\%$ , not $\frac{1}{9} \times 100\% \approx 11.1\%$ . The change is always compared with the original percentage.
Taking equal changes in points to be equal percentage changes. From 20% to 25% and from 50% to 55% are both rises of 5 percentage points, but they are increases of 25% and of 10%.
Reading "rose by 10%" as "add 10". In mathematics, a percentage of 40% that rises by 10% becomes 44%, since 10% of 40% is 4 percentage points. A rise of 10 percentage points would make it 50%.
Turning percentage points into numbers of people without the base. 3 percentage points of 500 students is 15 students, but 3 percentage points of 300 students is only 9 students.

Practice

Work each question on paper before you open its solution. For every change of a percentage, write down both percentages first.

Question 1

Find the change in percentage points, and say whether it is a rise or a fall.
(a) From 45% to 52%
(b) From 18.5% to 12%
(c) From 3% to 3.25%

ShowHide solution

(a) $52 - 45 = 7$ : a rise of 7 percentage points.

(b) $12 - 18.5 = -6.5$ : a fall of 6.5 percentage points.

(a) A rise of 7 percentage points
(b) A fall of 6.5 percentage points
(c) A rise of 0.25 percentage points

Question 2

In Term 1, 16% of the Secondary 1 students of a school got an A grade in mathematics. In Term 2, 22% got an A grade.
(a) Find the change in percentage points.
(b) Find the percentage change in the percentage of students with an A grade.

ShowHide solution

(a) $22 - 16 = 6$ : a rise of 6 percentage points.

(b) The original percentage is 16%. Percentage increase $= \frac{6}{16} \times 100\% = 37.5\%$

(a) A rise of 6 percentage points
(b) An increase of 37.5%

Question 3

A bank raises the yearly interest rate on a savings account from 2.5% to 2.8%. Describe the change
(a) in percentage points,
(b) as a percentage of the old rate.

ShowHide solution

(a) $2.8 - 2.5 = 0.3$ : a rise of 0.3 percentage points.

(b) The old rate is 2.5%. Percentage increase $= \frac{0.3}{2.5} \times 100\% = 12\%$

(a) The rate went up by 0.3 percentage points.
(b) The rate went up by 12%.

Question 4

The percentage of a school's students who take the school bus fell from 32% to 24%. Find the fall in percentage points and the percentage decrease.

ShowHide solution

Change in percentage points: $24 - 32 = -8$ , a fall of 8 percentage points.

The original percentage is 32%. Percentage decrease $= \frac{8}{32} \times 100\% = 25\%$

The percentage fell by 8 percentage points, a decrease of 25%.

Question 5

Last season Ravi scored with 40% of his shots at goal. This season he scored with 50% of his shots.
Ali says: "Ravi improved by 10%."
Siti says: "Ravi improved by 25%."
Mei Ling says: "Ravi improved by 10 percentage points."
Who is right? Correct the one who is wrong.

ShowHide solution

Change in percentage points: $50 - 40 = 10$ , so Mei Ling is right.

Percentage increase: $\frac{10}{40} \times 100\% = 25\%$ , so Siti is right.

Ali has written "%" for the change in a percentage. An improvement of 10% on 40% would be $40 + \frac{10}{100} \times 40 = 44$ , that is, 44%, not 50%.

Siti and Mei Ling are right. Ali should say "Ravi improved by 10 percentage points".

Question 6

At School P the pass rate of a test rose from 50% to 60%. At School Q it rose from 80% to 90%.
(a) Find the rise in percentage points for each school.
(b) Find the percentage increase in the pass rate for each school.
(c) The failure rate is the percentage of students who failed. Find the percentage decrease in the failure rate for each school.
(d) A newspaper says that the two schools "improved by the same amount". In what way is it right, and why might each school describe its result differently?

ShowHide solution

(a) School P: $60 - 50 = 10$ . School Q: $90 - 80 = 10$ . Both rose by 10 percentage points.

(b) School P: $\frac{10}{50} \times 100\% = 20\%$ . School Q: $\frac{10}{80} \times 100\% = 12.5\%$

(c) The pass rate and the failure rate add up to 100%. School P's failure rate went from 50% to 40%: a fall of 10 percentage points, and $\frac{10}{50} \times 100\% = 20\%$ .

School Q's failure rate went from 20% to 10%: a fall of 10 percentage points, and $\frac{10}{20} \times 100\% = 50\%$ . Its failure rate has halved.

(d) In percentage points both pass rates rose by 10, so the newspaper is right in that way. Compared with where each school started, the changes are different.

(a) 10 percentage points for each school
(b) School P 20%, School Q 12.5%
(c) School P 20%, School Q 50%
(d) Both rose by 10 percentage points. School P would rather say that its pass rate rose by 20%; School Q would rather say that its failure rate was halved.

Question 7

Mei Ling scored 60% in her first mathematics test. Her teacher writes: "In the second test her score rose by 10%."
(a) What are the two scores this could mean?
(b) Rewrite the sentence clearly for each.

ShowHide solution

(a) Reading 1, up 10 percentage points: $60 + 10 = 70$ , so 70%.

Reading 2, up 10% of 60%: $\frac{10}{100} \times 60 = 6$ percentage points, so $60 + 6 = 66$ , that is, 66%.

(b) Name the measure and give both percentages.

(a) 70% or 66%
(b) "Her score rose by 10 percentage points, from 60% to 70%" or "Her score rose by 10%, from 60% to 66%".

Question 8

A percentage is 40%.
(a) It rises by 25%. What is the new percentage, and by how many percentage points has it risen?
(b) Instead, it rises by 25 percentage points. What is the new percentage, and what is the percentage increase?

ShowHide solution

(a) 25% of 40% is $\frac{25}{100} \times 40 = 10$ percentage points. New percentage $= 40 + 10 = 50$ , so 50%.

It has risen by 10 percentage points.

(b) New percentage $= 40 + 25 = 65$ , so 65%.

Percentage increase $= \frac{25}{40} \times 100\% = 62.5\%$

(a) 50%, a rise of 10 percentage points
(b) 65%, an increase of 62.5%

Question 9

A school has 1200 students. In 2024, 15% of them borrowed a book from the school library each week. In 2025, 18% did, and the school still had 1200 students.
(a) Find the rise in percentage points.
(b) Find the percentage increase in the percentage of students who borrowed a book each week.
(c) How many more students borrowed a book each week in 2025? Express this as a percentage of the number in 2024, and compare it with your answer to (b).

ShowHide solution

(a) $18 - 15 = 3$ : a rise of 3 percentage points.

(b) Percentage increase $= \frac{3}{15} \times 100\% = 20\%$

$216 - 180 = 36$ , so 36 more students. Check: 1 percentage point of 1200 students is 12 students, and $3 \times 12 = 36$ .

As a percentage of the number in 2024: $\frac{36}{180} \times 100\% = 20\%$ . This is the same as (b), because the number of students in the school did not change.

(a) A rise of 3 percentage points
(b) 20%
(c) 36 more students, 20% more than in 2024: the same as (b), because the base (1200 students) stayed the same.

Question 10

A survey found that 12% of a school's students read for fun every day last year, and 15% this year. Three headlines were written:
A: "Daily readers up 25%"
B: "Daily readers up 3 points"
C: "Daily readers up 3%"
(a) Which headlines are right?
(b) Which right headline makes the change sound bigger?
(c) Write a headline that is clear and fair.

ShowHide solution

Change in percentage points: $15 - 12 = 3$ . Percentage increase: $\frac{3}{12} \times 100\% = 25\%$

(a) A is right. B is right: "3 points" means 3 percentage points, though it is clearer to say so. C is wrong: 3% of 12% is $\frac{3}{100} \times 12 = 0.36$ of a percentage point, which would give only 12.36%.

(b) A: "25%" sounds bigger than "3 points".

(a) A and B
(b) A
(c) For example: "Daily readers up from 12% to 15% of students".

Question 11

In last year's election for the chairperson of a school club, 40 members voted and Hui Min received 15 of the votes. This year 48 members voted and she received 24 of the votes.
(a) Find her share of the votes in each year, as a percentage of the members who voted.
(b) By how many percentage points did her share rise?
(c) Find the percentage increase in her share of the votes.
(d) Find the percentage increase in her number of votes. Why is it not the same as your answer to (c)?

ShowHide solution

(a) Last year: $\frac{15}{40} \times 100\% = 37.5\%$ . This year: $\frac{24}{48} \times 100\% = 50\%$ .

(b) $50 - 37.5 = 12.5$ : a rise of 12.5 percentage points.

(d) She received $24 - 15 = 9$ more votes. Percentage increase $= \frac{9}{15} \times 100\% = 60\%$

The base changed: 48 members voted this year and 40 last year. Her share this year is a percentage of more votes, so her number of votes grew faster than her share. Had her share stayed at 37.5%, she would still have received $\frac{37.5}{100} \times 48 = 18$ votes, 3 more than last year.

(a) 37.5% last year and 50% this year
(b) 12.5 percentage points
(c) About 33.3%
(d) 60%. The two shares are percentages of different numbers of votes (40 and 48), so the change in her share is not the change in her number of votes.

Question 12

The price of a bag before GST is $250.
(a) Find the GST on the bag at a rate of 8% and at a rate of 9%.
(b) By how much did the GST on the bag go up when the rate went from 8% to 9%? Give the answer in dollars and as a percentage of the GST at 8%.
(c) Find the price of the bag with GST at each rate, and the percentage increase in the price with GST, correct to 3 significant figures.
(d) Siti says: "When GST went up by 1 percentage point, the bag became 1% dearer." Is she right?

ShowHide solution

(a) At 8%: $\frac{8}{100} \times 250 = 20$ , so $20. At 9%: $\frac{9}{100} \times 250 = 22.5$ , so $22.50.

(b) $22.50 - 20 = 2.50$ , so the GST went up by $2.50. As a percentage of the GST at 8%: $\frac{2.50}{20} \times 100\% = 12.5\%$ . This is the same as the 12.5% rise in the rate, because the price before GST stayed the same.

Percentage increase $= \frac{2.50}{270} \times 100\% = 0.9259\ldots\% \approx 0.926\%$

(d) The extra $2.50 is 1% of the price before GST, $250. But the bag cost $270 with GST before the change, and $2.50 is only about 0.926% of $270.

(a) $20 at 8% and $22.50 at 9%
(b) $2.50, which is 12.5% of the GST at 8%
(c) $270 and $272.50; an increase of about 0.926%
(d) No. The price with GST rose by about 0.926%, not 1%. The extra $2.50 is 1% of the price before GST, but the bag already cost $270 with GST.

Key Points

When a percentage changes, the difference of the two percentages is measured in percentage points: subtract the original percentage from the new percentage.
The percentage change of a percentage compares that difference with the original percentage: $\frac{\text{rise or fall in percentage points}}{\text{original percentage}} \times 100\%$ .
A rate that goes from 8% to 9% rises by 1 percentage point and by 12.5%. Both are true; they measure different things.
The same number of percentage points is a bigger percentage change when the original percentage is smaller.
Never write "%" for the change in a percentage: from 6% to 8.5% is a rise of 2.5 percentage points, not 2.5%.
"Rose by 5%", said of a percentage, is unclear in everyday writing. Write "5 percentage points" if that is meant, or give both percentages ("from 50% to 55%").
Each percentage point is 1% of the base. When the base stays the same, the percentage change of a rate is also the percentage change of the amount it gives.
A writer may choose whichever measure sounds bigger or smaller. A fair statement gives both percentages.