SAT Growth Rate vs Growth Factor: Stop Mixing Up 6% and 1.06

Learn the difference between exponential growth rate, decay rate, and growth factor on the SAT, including repeated percent change, tables, equations, and common traps.

Reviewed by Aakash S. · Cornell University '14 · Counselling Advisor, World ViralPublished 29 September 2026Last updated 29 September 2026

If a quantity grows by 6% each period:

the growth rate is $6\%=0.06$ ;
the growth factor is $1.06$ .

Those are related, but they are not interchangeable.

For exponential growth:

A(t)=A_0(1+r)^t,

where:

$A_0$ is the initial amount;
$r$ is the growth rate written as a decimal;
$1+r$ is the growth factor.

For exponential decay:

A(t)=A_0(1-r)^t.

A large share of SAT percentage-model mistakes come from confusing $r$ with $1+r$ or $1-r$ .

Current SAT connection: growth factor and percent change

This topic sits at the intersection of SAT exponential functions and percentage reasoning. College Board's current specifications include exponential modeling in Advanced Math and percentage/proportional reasoning in Problem-Solving and Data Analysis.

Use SAT Exponential Growth and Decay for the parent model, Linear vs Exponential Models for model identification, and SAT Percentages for the broader percent toolkit.

Why the factor includes 1

Suppose you begin with 100.

A 6% increase is:

6\%\text{ of }100=6.

New amount:

100+6=106.

That is:

106\%=1.06

of the original.

So multiplying by:

1.06

does both jobs at once:

keeps the original 100%;
adds another 6%.

That is why the factor is not 0.06.

Growth-rate conversion table

Percent growth	Rate $r$	Growth factor $1+r$
2%	0.02	1.02
6%	0.06	1.06
15%	0.15	1.15
40%	0.40	1.40
125%	1.25	2.25

The last row is important.

A 125% increase means the new amount is:

225\%

of the original.

So the factor is:

2.25.

Decay-rate conversion table

Percent decrease	Rate $r$	Decay factor $1-r$
5%	0.05	0.95
18%	0.18	0.82
30%	0.30	0.70
72%	0.72	0.28

A decrease by 18% means the quantity retains 82%.

So multiply by:

0.82.

Worked example 1: write a growth model

A population begins at 24,000 and grows by 4% each year.

Initial amount:

A_0=24000.

Growth rate:

r=0.04.

Growth factor:

1+r=1.04.

Model:

\boxed{P(t)=24000(1.04)^t}.

Worked example 2: write a decay model

A laptop costs $1,500 and loses 22% of its value each year.

Decay rate:

0.22.

Retained fraction:

1-0.22=0.78.

Model:

\boxed{V(t)=1500(0.78)^t}.

Do not use:

1500(0.22)^t.

That would mean 78% is lost immediately each year and only 22% remains.

How to read the rate from an equation

Suppose:

A(t)=900(1.08)^t.

The growth factor is:

1.08.

Subtract 1:

1.08-1=0.08.

Convert to percent:

8\%.

So the quantity grows by 8% per period.

How to read decay from an equation

Suppose:

A(t)=900(0.73)^t.

The factor 0.73 means 73% remains after each period.

So percent lost is:

100\%-73\%=27\%.

Decay rate:

27\%.

Worked example 3: SAT wording trap

A town's population is modeled by:

P(t)=18000(1.035)^t.

Which statement is correct?

The factor is:

1.035.

Growth rate:

1.035-1=0.035=3.5\%.

So the population increases by:

\boxed{3.5\%\text{ per period}}.

It does not increase by 103.5%.

103.5% is the percentage of the previous amount retained after growth, not the increase itself.

Factor from a table

Suppose:

$t$	$A(t)$
0	500
1	575
2	661.25

Compute ratio:

\frac{575}{500}=1.15.

Check next:

\frac{661.25}{575}=1.15.

Growth factor:

1.15.

Growth rate:

15\%.

Model:

A(t)=500(1.15)^t.

Worked example 4: decay from a table

$t$	$A(t)$
0	800
1	680
2	578

Ratio:

\frac{680}{800}=0.85.

\frac{578}{680}=0.85.

Decay factor:

0.85.

So 85% remains.

Decay rate:

15\%.

Repeated percent change compounds

Suppose $1,000 grows by 10% per year.

After one year:

1000(1.10)=1100.

After two:

1100(1.10)=1210.

Equivalent:

1000(1.10)^2=1210.

The second year's 10% is:

110,

not 100.

Why?

Because it is 10% of the new balance.

That is compounding.

Why 10% twice is not simply 20%

Two 10% increases produce:

1.10^2=1.21.

That is a 21% total increase.

The extra 1% comes from growth on the first period's growth.

For repeated changes, multiply factors rather than add percentages.

Worked example 5: increase then decrease

A price increases 20% and then decreases 20%.

Start with 100.

After increase:

100(1.20)=120.

After decrease:

120(0.80)=96.

Final amount:

96.

Overall change:

-4\%.

Why not zero?

Because the 20% decrease is calculated from 120, not the original 100.

Factor method:

1.20(0.80)=0.96.

Reverse-engineering a growth factor

Suppose a quantity changes from 250 to 290 in one period.

Factor:

\frac{290}{250}=1.16.

Growth rate:

1.16-1=0.16.

So:

\boxed{16\%}.

This ratio method works whenever one period's old and new values are known.

Reverse-engineering a decay factor

A quantity changes from 500 to 425.

Factor:

\frac{425}{500}=0.85.

So 85% remains.

Decay rate:

15\%.

Multi-period growth factor

Suppose a quantity grows by 5% per month for 6 months.

Monthly factor:

1.05.

Six-month factor:

(1.05)^6.

Total percent increase is:

\left((1.05)^6-1\right)\times100\%.

Do not multiply 5% by 6 unless the problem explicitly describes a linear approximation.

Worked example 6: infer annual factor from two-year change

A population increases from 10,000 to 12,100 over 2 years at a constant annual percentage rate.

Let annual factor be:

b.

Then:

10000b^2=12100.

b^2=1.21.

b=1.10.

Annual growth rate:

10\%.

The two-year total increase is 21%, but the annual rate is 10%.

Doubling and halving

If a quantity doubles each period, factor:

2.

Growth rate:

100\%.

If a quantity halves each period, factor:

0.5.

Decay rate:

50\%.

“Doubles” does not mean factor $1.2$ .

“Halves” does not mean subtract 0.5 units.

Translate words into multiplicative factors.

“Increases by 150%” vs “becomes 150%”

These are different.

Increases by 150%

New value:

100\%+150\%=250\%

of original.

Factor:

2.5.

Becomes 150% of original

Factor:

1.5.

Increase:

50\%.

This wording distinction appears frequently in percentage reasoning.

Worked example 7: percent more than

A value $B$ is 30% greater than $A$ .

Then:

B=A(1.30).

So factor from $A$ to $B$ :

1.30.

But $A$ is not 30% less than $B$ .

If:

B=1.30A,

then:

A=\frac{B}{1.30}\approx0.7692B.

So $A$ is about 23.08% less than $B$ .

Percent comparisons depend on the reference value.

Growth factor with intervals longer than one unit

Suppose a quantity triples every 4 years.

If $t$ is measured in years:

A(t)=A_0(3)^{t/4}.

The exponent:

\frac{t}{4}

counts how many 4-year periods have passed.

If you instead use $n$ for number of 4-year periods:

A(n)=A_0(3)^n.

Worked example 8: convert a two-year factor to yearly factor

A quantity multiplies by 1.44 every 2 years.

If annual growth factor is $b$ :

b^2=1.44.

b=1.2.

So annual growth rate:

20\%.

Do not say 44% per year.

44% is the two-year total increase.

Common trap 1: base equals rate

Wrong for 7% growth:

A_0(0.07)^t.

Correct:

A_0(1.07)^t.

Common trap 2: decay factor equals amount lost

Wrong for 18% decay:

0.18.

Correct retained factor:

0.82.

Common trap 3: reading 0.92 as 92% decay

0.92

means 92% remains.

Decay is:

8\%.

Common trap 4: adding repeated percentages

Repeated percentage changes multiply.

Two 25% increases:

1.25^2=1.5625.

Overall increase:

56.25\%.

Not 50%.

Common trap 5: ignoring the period

5% per month and 5% per year are very different.

The exponent must count the correct number of periods.

Common trap 6: confusing factor with percent of original

Factor:

1.30

means new amount is 130% of original, which is a 30% increase.

Common trap 7: reversing percent relationships

If B is 20% greater than A:

B=1.2A.

It does not follow that:

A=0.8B.

Instead:

A=\frac{B}{1.2}.

A four-step translation method

Whenever you see repeated percent change:

Convert the percent to decimal $r$ .
Decide growth or decay.
Build the factor:
- growth: $1+r$ ;
- decay: $1-r$ .
Raise that factor to the number of periods.

Example: 9% annual decay for $t$ years:

A(t)=A_0(0.91)^t.

Mini drill

1. 12% growth

Rate:

0.12.

Factor:

1.12.

2. 12% decay

Rate:

0.12.

Factor:

0.88.

3. Factor 1.27

Growth rate:

27\%.

4. Factor 0.64

Decay rate:

36\%.

5. Increase by 80%

Factor:

1.80.

6. Becomes 80% of original

Factor:

0.80.

Decay:

20\%.

7. Doubles

Factor:

2.

Growth rate:

100\%.

The deeper lesson

A percentage rate describes change relative to the old amount.

A factor describes the new amount relative to the old amount.

That is why:

growth factor=1+growth rate

and:

\text{decay factor}=1-\text{decay rate}.

Once you separate those two ideas, exponential models become much easier to interpret.

FAQ: SAT growth rate vs growth factor

What is the growth factor for a 6% increase?

1+0.06=1.06.

What is the decay factor for a 6% decrease?

1-0.06=0.94.

Why isn't the base 0.06 for 6% growth?

Because 0.06 represents only the added portion. The new amount keeps the original 100% plus another 6%, for a total factor of 1.06.

What does a factor of 1.25 mean?

A 25% increase per period.

What does a factor of 0.72 mean?

The quantity retains 72% each period, so it decreases by 28%.

Why don't repeated percentage changes add?

Each percentage applies to a changing base. Multiply the period factors instead.

Is a 100% increase the same as doubling?

Yes. A 100% increase gives factor 2.

Final takeaway

For repeated percentage change:

\boxed{\text{growth: }A=A_0(1+r)^t}

\boxed{\text{decay: }A=A_0(1-r)^t}

Remember:

6% growth → rate $0.06$ , factor $1.06$ ;
6% decay → rate $0.06$ , factor $0.94$ .

Most mistakes happen because students put the rate where the model needs the factor.

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SAT Growth Rate vs Growth Factor: Stop Mixing Up 6% and 1.06

Learn the difference between exponential growth rate, decay rate, and growth factor on the SAT, including repeated percent change, tables, equations, and common traps.

Reviewed by Aakash S. · Cornell University '14 · Counselling Advisor, World ViralPublished 29 September 2026Last updated 29 September 2026

If a quantity grows by 6% each period:

the growth rate is $6\%=0.06$ ;
the growth factor is $1.06$ .

Those are related, but they are not interchangeable.

For exponential growth:

A(t)=A_0(1+r)^t,

where:

$A_0$ is the initial amount;
$r$ is the growth rate written as a decimal;
$1+r$ is the growth factor.

For exponential decay:

A(t)=A_0(1-r)^t.

A large share of SAT percentage-model mistakes come from confusing $r$ with $1+r$ or $1-r$ .

Current SAT connection: growth factor and percent change

Use SAT Exponential Growth and Decay for the parent model, Linear vs Exponential Models for model identification, and SAT Percentages for the broader percent toolkit.

Why the factor includes 1

Suppose you begin with 100.

A 6% increase is:

6\%\text{ of }100=6.

New amount:

100+6=106.

That is:

106\%=1.06

of the original.

So multiplying by:

1.06

does both jobs at once:

keeps the original 100%;
adds another 6%.

That is why the factor is not 0.06.

Growth-rate conversion table

Percent growth	Rate $r$	Growth factor $1+r$
2%	0.02	1.02
6%	0.06	1.06
15%	0.15	1.15
40%	0.40	1.40
125%	1.25	2.25

The last row is important.

A 125% increase means the new amount is:

225\%

of the original.

So the factor is:

2.25.

Decay-rate conversion table

Percent decrease	Rate $r$	Decay factor $1-r$
5%	0.05	0.95
18%	0.18	0.82
30%	0.30	0.70
72%	0.72	0.28

A decrease by 18% means the quantity retains 82%.

So multiply by:

0.82.

Worked example 1: write a growth model

A population begins at 24,000 and grows by 4% each year.

Initial amount:

A_0=24000.

Growth rate:

r=0.04.

Growth factor:

1+r=1.04.

Model:

\boxed{P(t)=24000(1.04)^t}.

Worked example 2: write a decay model

A laptop costs $1,500 and loses 22% of its value each year.

Decay rate:

0.22.

Retained fraction:

1-0.22=0.78.

Model:

\boxed{V(t)=1500(0.78)^t}.

Do not use:

1500(0.22)^t.

That would mean 78% is lost immediately each year and only 22% remains.

How to read the rate from an equation

Suppose:

A(t)=900(1.08)^t.

The growth factor is:

1.08.

Subtract 1:

1.08-1=0.08.

Convert to percent:

8\%.

So the quantity grows by 8% per period.

How to read decay from an equation

Suppose:

A(t)=900(0.73)^t.

The factor 0.73 means 73% remains after each period.

So percent lost is:

100\%-73\%=27\%.

Decay rate:

27\%.

Worked example 3: SAT wording trap

A town's population is modeled by:

P(t)=18000(1.035)^t.

Which statement is correct?

The factor is:

1.035.

Growth rate:

1.035-1=0.035=3.5\%.

So the population increases by:

\boxed{3.5\%\text{ per period}}.

It does not increase by 103.5%.

103.5% is the percentage of the previous amount retained after growth, not the increase itself.

Factor from a table

Suppose:

$t$	$A(t)$
0	500
1	575
2	661.25

Compute ratio:

\frac{575}{500}=1.15.

Check next:

\frac{661.25}{575}=1.15.

Growth factor:

1.15.

Growth rate:

15\%.

Model:

A(t)=500(1.15)^t.

Worked example 4: decay from a table

$t$	$A(t)$
0	800
1	680
2	578

Ratio:

\frac{680}{800}=0.85.

\frac{578}{680}=0.85.

Decay factor:

0.85.

So 85% remains.

Decay rate:

15\%.

Repeated percent change compounds

Suppose $1,000 grows by 10% per year.

After one year:

1000(1.10)=1100.

After two:

1100(1.10)=1210.

Equivalent:

1000(1.10)^2=1210.

The second year's 10% is:

110,

not 100.

Why?

Because it is 10% of the new balance.

That is compounding.

Why 10% twice is not simply 20%

Two 10% increases produce:

1.10^2=1.21.

That is a 21% total increase.

The extra 1% comes from growth on the first period's growth.

For repeated changes, multiply factors rather than add percentages.

Worked example 5: increase then decrease

A price increases 20% and then decreases 20%.

Start with 100.

After increase:

100(1.20)=120.

After decrease:

120(0.80)=96.

Final amount:

96.

Overall change:

-4\%.

Why not zero?

Because the 20% decrease is calculated from 120, not the original 100.

Factor method:

1.20(0.80)=0.96.

Reverse-engineering a growth factor

Suppose a quantity changes from 250 to 290 in one period.

Factor:

\frac{290}{250}=1.16.

Growth rate:

1.16-1=0.16.

So:

\boxed{16\%}.

This ratio method works whenever one period's old and new values are known.

Reverse-engineering a decay factor

A quantity changes from 500 to 425.

Factor:

\frac{425}{500}=0.85.

So 85% remains.

Decay rate:

15\%.

Multi-period growth factor

Suppose a quantity grows by 5% per month for 6 months.

Monthly factor:

1.05.

Six-month factor:

(1.05)^6.

Total percent increase is:

\left((1.05)^6-1\right)\times100\%.

Do not multiply 5% by 6 unless the problem explicitly describes a linear approximation.

Worked example 6: infer annual factor from two-year change

A population increases from 10,000 to 12,100 over 2 years at a constant annual percentage rate.

Let annual factor be:

b.

Then:

10000b^2=12100.

b^2=1.21.

b=1.10.

Annual growth rate:

10\%.

The two-year total increase is 21%, but the annual rate is 10%.

Doubling and halving

If a quantity doubles each period, factor:

2.

Growth rate:

100\%.

If a quantity halves each period, factor:

0.5.

Decay rate:

50\%.

“Doubles” does not mean factor $1.2$ .

“Halves” does not mean subtract 0.5 units.

Translate words into multiplicative factors.

“Increases by 150%” vs “becomes 150%”

These are different.

Increases by 150%

New value:

100\%+150\%=250\%

of original.

Factor:

2.5.

Becomes 150% of original

Factor:

1.5.

Increase:

50\%.

This wording distinction appears frequently in percentage reasoning.

Worked example 7: percent more than

A value $B$ is 30% greater than $A$ .

Then:

B=A(1.30).

So factor from $A$ to $B$ :

1.30.

But $A$ is not 30% less than $B$ .

If:

B=1.30A,

then:

A=\frac{B}{1.30}\approx0.7692B.

So $A$ is about 23.08% less than $B$ .

Percent comparisons depend on the reference value.

Growth factor with intervals longer than one unit

Suppose a quantity triples every 4 years.

If $t$ is measured in years:

A(t)=A_0(3)^{t/4}.

The exponent:

\frac{t}{4}

counts how many 4-year periods have passed.

If you instead use $n$ for number of 4-year periods:

A(n)=A_0(3)^n.

Worked example 8: convert a two-year factor to yearly factor

A quantity multiplies by 1.44 every 2 years.

If annual growth factor is $b$ :

b^2=1.44.

b=1.2.

So annual growth rate:

20\%.

Do not say 44% per year.

44% is the two-year total increase.

Common trap 1: base equals rate

Wrong for 7% growth:

A_0(0.07)^t.

Correct:

A_0(1.07)^t.

Common trap 2: decay factor equals amount lost

Wrong for 18% decay:

0.18.

Correct retained factor:

0.82.

Common trap 3: reading 0.92 as 92% decay

0.92

means 92% remains.

Decay is:

8\%.

Common trap 4: adding repeated percentages

Repeated percentage changes multiply.

Two 25% increases:

1.25^2=1.5625.

Overall increase:

56.25\%.

Not 50%.

Common trap 5: ignoring the period

5% per month and 5% per year are very different.

The exponent must count the correct number of periods.

Common trap 6: confusing factor with percent of original

Factor:

1.30

means new amount is 130% of original, which is a 30% increase.

Common trap 7: reversing percent relationships

If B is 20% greater than A:

B=1.2A.

It does not follow that:

A=0.8B.

Instead:

A=\frac{B}{1.2}.

A four-step translation method

Whenever you see repeated percent change:

Convert the percent to decimal $r$ .
Decide growth or decay.
Build the factor:
- growth: $1+r$ ;
- decay: $1-r$ .
Raise that factor to the number of periods.

Example: 9% annual decay for $t$ years:

A(t)=A_0(0.91)^t.

Mini drill

1. 12% growth

Rate:

0.12.

Factor:

1.12.

2. 12% decay

Rate:

0.12.

Factor:

0.88.

3. Factor 1.27

Growth rate:

27\%.

4. Factor 0.64

Decay rate:

36\%.

5. Increase by 80%

Factor:

1.80.

6. Becomes 80% of original

Factor:

0.80.

Decay:

20\%.

7. Doubles

Factor:

2.

Growth rate:

100\%.

The deeper lesson

A percentage rate describes change relative to the old amount.

A factor describes the new amount relative to the old amount.

That is why:

growth factor=1+growth rate

and:

\text{decay factor}=1-\text{decay rate}.

Once you separate those two ideas, exponential models become much easier to interpret.

FAQ: SAT growth rate vs growth factor

What is the growth factor for a 6% increase?

1+0.06=1.06.

What is the decay factor for a 6% decrease?

1-0.06=0.94.

Why isn't the base 0.06 for 6% growth?

Because 0.06 represents only the added portion. The new amount keeps the original 100% plus another 6%, for a total factor of 1.06.

What does a factor of 1.25 mean?

A 25% increase per period.

What does a factor of 0.72 mean?

The quantity retains 72% each period, so it decreases by 28%.

Why don't repeated percentage changes add?

Each percentage applies to a changing base. Multiply the period factors instead.

Is a 100% increase the same as doubling?

Yes. A 100% increase gives factor 2.

Final takeaway

For repeated percentage change:

\boxed{\text{growth: }A=A_0(1+r)^t}

\boxed{\text{decay: }A=A_0(1-r)^t}

Remember:

6% growth → rate $0.06$ , factor $1.06$ ;
6% decay → rate $0.06$ , factor $0.94$ .

Most mistakes happen because students put the rate where the model needs the factor.

How to practice this on MastaPrep

See how MastaPrep works or start with free SAT practice questions. No credit card is needed for the free plan.

Trademark note: SAT® is a registered trademark of College Board. MastaPrep is independent and is not affiliated with or endorsed by College Board.

Ready to put this into practice?

Practice SAT Math free

Free to start. No credit card.

Current SAT connection: growth factor and percent change

Why the factor includes 1

Growth-rate conversion table

Decay-rate conversion table

Worked example 1: write a growth model

Worked example 2: write a decay model

How to read the rate from an equation

How to read decay from an equation

Worked example 3: SAT wording trap

Factor from a table

Worked example 4: decay from a table

Repeated percent change compounds

Why 10% twice is not simply 20%

Worked example 5: increase then decrease

Reverse-engineering a growth factor

Reverse-engineering a decay factor

Multi-period growth factor

Worked example 6: infer annual factor from two-year change

Doubling and halving

“Increases by 150%” vs “becomes 150%”

Increases by 150%

Becomes 150% of original

Worked example 7: percent more than

Growth factor with intervals longer than one unit

Worked example 8: convert a two-year factor to yearly factor

Common trap 1: base equals rate

Common trap 2: decay factor equals amount lost

Common trap 3: reading 0.92 as 92% decay

Common trap 4: adding repeated percentages

Common trap 5: ignoring the period

Common trap 6: confusing factor with percent of original

Common trap 7: reversing percent relationships

A four-step translation method

Mini drill

1. 12% growth

2. 12% decay

3. Factor 1.27

4. Factor 0.64

5. Increase by 80%

6. Becomes 80% of original

7. Doubles

The deeper lesson

FAQ: SAT growth rate vs growth factor

What is the growth factor for a 6% increase?

What is the decay factor for a 6% decrease?

Why isn't the base 0.06 for 6% growth?

What does a factor of 1.25 mean?

What does a factor of 0.72 mean?

Why don't repeated percentage changes add?

Is a 100% increase the same as doubling?

Final takeaway

How to practice this on MastaPrep

Related MastaPrep guides

Read next

Current SAT connection: growth factor and percent change

Why the factor includes 1

Growth-rate conversion table

Decay-rate conversion table

Worked example 1: write a growth model

Worked example 2: write a decay model

How to read the rate from an equation

How to read decay from an equation

Worked example 3: SAT wording trap

Factor from a table

Worked example 4: decay from a table

Repeated percent change compounds

Why 10% twice is not simply 20%

Worked example 5: increase then decrease

Reverse-engineering a growth factor

Reverse-engineering a decay factor

Multi-period growth factor

Worked example 6: infer annual factor from two-year change

Doubling and halving

“Increases by 150%” vs “becomes 150%”

Increases by 150%

Becomes 150% of original

Worked example 7: percent more than

Growth factor with intervals longer than one unit

Worked example 8: convert a two-year factor to yearly factor

Common trap 1: base equals rate