Two equivalent expressions always have the same value, but they can tell different stories. Rewriting an expression can make its meaning easier to see, especially with percent change.
Key idea
$a + 0.05a = 1.05a$: adding 5% of an amount is the same as multiplying it by $1.05$. Likewise $p - 0.2p = 0.8p$: taking 20% off leaves 80%. Choose the form that shows what you need to see.
Why it works
It is the distributive property again: $a + 0.05a = 1a + 0.05a = (1 + 0.05)a = 1.05a$. The first form shows the price and the increase separately; the second shows the new price as one multiplication.
Worked examples
Worked example: An increase
Write $a + 0.05a$ as a single term. What does it mean?
- $a + 0.05a = (1 + 0.05)a = 1.05a$.
- Meaning: the new amount is 105% of $a$, a 5% increase.
Answer: $1.05a$
Worked example: A discount
A store marks every price $p$ down to $p - 0.2p$. Rewrite it and describe the sale.
- $p - 0.2p = (1 - 0.2)p = 0.8p$.
- You pay 80% of the price: the sale is 20% off.
Answer: $0.8p$, which is 20% off
Worked example: Two stories, one total
Four friends each buy a ticket for $x$ dollars and a \$3 snack. Show that $4(x + 3)$ and $4x + 12$ both give the total.
- $4(x + 3)$: each person spends $x + 3$, and there are 4 people.
- $4x + 12$: the tickets cost $4x$ in all and the snacks cost $4 \times 3 = 12$.
- They are equal by the distributive property.
Answer: Both are the total.
Common mistake
Reading a percent as a plain number: a 6% increase on $p$ is $p + 0.06p = 1.06p$, not $p + 6$. And $1.6p$ would be a 60% increase. Check with $p = 100$: a 6% increase should give 106.