Decimals and Percents

💯 Decimals and Percents

Part 1 of 5 — Place Value & What a Decimal Really Means

Topics in This Part

🔑 Key Concept: A decimal is a shorthand for a fraction whose denominator is a power of ten. Once you see which power of ten each place stands for, every decimal skill follows.

Reading the Places After the Point

Each spot to the right of the decimal point is ten times smaller than the one before it.

So $0.475$ means $\dfrac{4}{10} + \dfrac{7}{100} + \dfrac{5}{1000} = \dfrac{475}{1000}$.

💡 Read it out loud: the last place names the whole thing. $0.475$ is read "four hundred seventy-five thousandths" because the final digit $5$ sits in the thousandths place.

Concept Check 🎯

Decimals Are Just Fractions in Disguise

To turn a decimal into a fraction, the number of decimal places tells you the denominator:

$0.7 = \frac{7}{10}, \qquad 0.07 = \frac{7}{100}, \qquad 0.007 = \frac{7}{1000}$

• 1 decimal place → denominator $10$
• 2 decimal places → denominator $100$
• 3 decimal places → denominator $1000$

Worked Example: write $0.24$ as a fraction in lowest terms

$0.24 = \frac{24}{100} = \frac{24 \div 4}{100 \div 4} = \frac{6}{25}$

🔑 Key Idea: Count the decimal places to get the denominator, drop the point to get the numerator, then simplify.

Decimal → Fraction 🧮

Write each decimal as a fraction in lowest terms. Type it like 3/4.

1) $0.5 = \,?$ 2) $0.35 = \,?$ 3) $0.08 = \,?$

Comparing & Ordering Decimals

To compare decimals, line up the decimal points and compare place by place, left to right. If one number has fewer digits, pad it with trailing zeros — they don't change its value.

Example: which is larger, $0.4$ or $0.38$?

Pad $0.4$ to $0.40$ and compare:

$0.40 \;\text{vs}\; 0.38 \;\Rightarrow\; \text{tenths } 4 > 3 \;\Rightarrow\; 0.4 > 0.38$

⚠️ Common trap: "$0.38$ has more digits, so it's bigger." Not true! More digits does not mean a bigger number. Compare place value, not length.

Compare the Decimals 🔽

Choose the correct symbol for each pair.

💯 Decimals and Percents

Part 2 of 5 — Arithmetic with Decimals

🔑 The Idea: Adding and subtracting decimals is all about lining up the points. Multiplying and dividing is about counting decimal places. Master those two habits and decimal arithmetic becomes routine.

Adding & Subtracting

Line up the decimal points, pad with zeros so every number has the same number of places, then add or subtract as usual — keeping the point in the same column.

Worked Example: $3.4 + 12.75$

$\begin{aligned} 3.40 \\ +\;12.75 \\ \hline 16.15 \end{aligned}$

Worked Example: $8 - 2.6$

Write $8$ as $8.0$, then subtract:

$8.0 - 2.6 = 5.4$

⚠️ Don't right-align the digits like whole numbers — align the decimal points. $3.4 + 12.75$ is not $3.4$ stacked under $2.75$.

Add & Subtract 🧮

1) $5.6 + 2.85 = \,?$ 2) $10 - 3.7 = \,?$ 3) $0.45 + 0.9 = \,?$

Multiplying Decimals

Ignore the points, multiply as whole numbers, then count the total decimal places in both factors and place the point that many spots from the right.

Worked Example: $0.6 \times 0.4$

$6 \times 4 = 24$. The factors have $1 + 1 = 2$ decimal places total, so:

$0.6 \times 0.4 = 0.24$

Worked Example: $1.2 \times 0.05$

$12 \times 5 = 60$. Total decimal places $= 1 + 2 = 3$, so move the point $3$ spots:

$1.2 \times 0.05 = 0.060 = 0.06$

💡 Sanity check: multiplying by a number less than 1 makes things smaller. $0.6 \times 0.4 = 0.24$ is smaller than both factors — that's expected.

Concept Check 🎯

Dividing Decimals

To divide by a decimal, move the divisor's point to the right until it's a whole number, and move the dividend's point the same number of places. Then divide normally.

Worked Example: $4.5 \div 0.9$

Multiply both by $10$ so the divisor becomes a whole number:

$4.5 \div 0.9 = 45 \div 9 = 5$

Worked Example: $1.44 \div 1.2$

Multiply both by $10$: $\;14.4 \div 12 = 1.2$.

🔑 Why it works: moving both points the same way multiplies top and bottom of $\dfrac{4.5}{0.9}$ by the same power of ten — the value of the fraction doesn't change.

Multiply & Divide 🧮

1) $0.8 \times 0.3 = \,?$ 2) $6 \div 0.5 = \,?$ 3) $0.96 \div 0.6 = \,?$

💯 Decimals and Percents

Part 3 of 5 — What a Percent Is & Converting

🔑 The Big Idea: Percent means "per hundred." A percent is just a fraction with a hidden denominator of $100$ — and that makes converting between percents, decimals, and fractions almost mechanical.

What Does "Percent" Mean?

The word percent comes from per centum — "per hundred." The symbol $\%$ literally stands for $\div 100$:

$37\% = \frac{37}{100} = 0.37$

So a percent is three views of the same number:

💡 A percent can be more than $100$. $150\%$ just means $1.5$ — one and a half of the whole.

Percent ↔ Decimal

Because $\%$ means $\div 100$, converting is just moving the decimal point two places:

• Percent → Decimal: move the point left 2 places (divide by 100). $45\% \to 0.45, \qquad 8\% \to 0.08, \qquad 120\% \to 1.20$
• Decimal → Percent: move the point right 2 places (multiply by 100). $0.6 \to 60\%, \qquad 0.03 \to 3\%, \qquad 1.25 \to 125\%$

⚠️ Watch the zeros: $8\% = 0.08$, not $0.8$. Moving left two places from $8.$ gives $0.08$. And $0.6 = 60\%$, not $6\%$ — pad to $0.60$ before moving.

Percent ↔ Decimal 🔽

Percent ↔ Fraction

• Percent → Fraction: put the percent over $100$, then simplify. $40\% = \frac{40}{100} = \frac{2}{5}$
• Fraction → Percent: rewrite with denominator $100$ (or divide and multiply by $100$). $\frac{3}{4} = \frac{75}{100} = 75\%$

A few worth memorizing

🔑 Knowing these "benchmark" conversions by heart makes estimating percents in your head fast.

Convert 🧮

1) Write $80\%$ as a fraction in lowest terms (like 3/4). 2) Write $\dfrac{1}{5}$ as a percent. (include the number only, e.g. 20) 3) Write $0.125$ as a percent. (number only)

Concept Check 🎯

💯 Decimals and Percents

Part 4 of 5 — Finding a Percent of a Number

🔑 The Move: "Percent of" almost always means multiply. Turn the percent into a decimal and multiply by the number. That one habit solves tips, discounts, and tax.

Finding a Percent OF a Number

To find a percent of a quantity, convert the percent to a decimal and multiply:

$\text{percent of a number} = (\text{decimal form}) \times (\text{number})$

Worked Example: What is $20\%$ of $60$?

$20\% = 0.20, \qquad 0.20 \times 60 = 12$

Worked Example: What is $8\%$ of $250$?

$8\% = 0.08, \qquad 0.08 \times 250 = 20$

💡 Quick mental trick: $10\%$ of any number is just that number with the point moved one place left. $10\%$ of $60$ is $6$, so $20\%$ is double that — $12$. ✓

Percent of a Number 🧮

1) $25\%$ of $80 = \,?$ 2) $10\%$ of $45 = \,?$ 3) $5\%$ of $200 = \,?$

Tips, Tax, and Discounts

These are all just "percent of a number," then add or subtract.

Example — Tip: a meal costs $40 and you tip $15\%$.

$0.15 \times 40 = 6 \;\Rightarrow\; \text{total} = 40 + 6 = 46$, giving $46

Example — Discount: a $50 jacket is $30\%$ off.

$0.30 \times 50 = 15 \;\Rightarrow\; \text{sale price} = 50 - 15 = 35$, giving $35

🔑 Shortcut for discounts: $30\%$ off means you pay $70\%$. So $0.70 \times 50 = 35$ giving $35 in a single step — same answer, fewer moves.

Real-World Percents 🔽

A $60 pair of shoes is $25\%$ off. Walk through it.

Concept Check 🎯

💯 Decimals and Percents

Part 5 of 5 — Percent Change & Mastery Check

You can now read decimals, do decimal arithmetic, convert between forms, and take a percent of a number. The last skill ties it together: measuring how much something changed as a percent.

Percent Increase & Decrease

To find the percent a quantity changed, compare the change to the original:

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

Worked Example — Increase: a price rises from $40 to $50.

$\frac{50 - 40}{40} \times 100\% = \frac{10}{40} \times 100\% = 25\% \text{ increase}$

Worked Example — Decrease: a class shrinks from $30$ to $24$ students.

$\frac{24 - 30}{30} \times 100\% = \frac{-6}{30} \times 100\% = -20\%,\; \text{a } 20\% \text{ decrease}$

⚠️ Always divide by the original amount, not the new one. The starting value is your "$100\%$."

Percent Change 🧮

Give the percent change (number only, e.g. 25).

1) A plant grows from $20$ cm to $26$ cm. Percent increase $= \,?$ 2) A $80 coat drops to $60. Percent decrease $= \,?$

Quick Reference

💡 Almost every percent problem reduces to one idea: percent means "per hundred." Convert to a decimal and the arithmetic is ordinary multiplication.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.