Arithmetic Review

What You’ll Learn

In this review lesson you’ll revisit the main ideas from the entire Arithmetic section: place value, whole number operations, fractions, decimals, and percentages. Work through the refreshers and mixed problems to solidify everything before moving on to algebra.

Place Value & Rounding

Every digit in a number has a value based on its position. In 4,827: the 4 is in the thousands place (worth 4,000), the 8 is hundreds (800), the 2 is tens (20), and the 7 is ones (7).

After the decimal point, positions go tenths, hundredths, thousandths. In 3.456: the 4 is tenths (0.4), the 5 is hundredths (0.05), the 6 is thousandths (0.006).

To round, look at the digit to the right of the place you’re rounding to. If it’s 5 or more, round up. Less than 5, keep it.

• Round $7{,}849$ to the nearest hundred: look at tens (4 < 5) → $7{,}800$
• Round $3.678$ to the nearest tenth: look at hundredths (7 ≥ 5) → $3.7$

Whole Number Operations

The four operations build on each other:

Addition & Subtraction - line up by place value, carry or borrow as needed.

$$\begin{array}{rrrr} & \scriptsize{1} & & \\ & 4 & 5 & 6 \\ {+} & 3 & 8 & 9 \\ \hline & 8 & 4 & 5 \end{array}$$

Multiplication - multiply by each digit, shift partial products, add them up.

$$\begin{array}{rrrr} & & \scriptsize{2} & \\ & & 3 & 4 \\ \times & & 1 & 2 \\ \hline & & 6 & 8 \\ & 3 & 4 & 0 \\ \hline & 4 & 0 & 8 \end{array}$$

Division - divide, multiply, subtract, bring down. Repeat.

$$\begin{array}{r} 23 \\ 7\overline{)161} \\ \underline{-14}\phantom{1} \\ 21 \\ \underline{-21} \\ 0 \end{array}$$

$161 \div 7 = 23$. Check: $23 \times 7 = 161$ ✓

Fractions

A fraction represents parts of a whole. The numerator (top) counts the parts, the denominator (bottom) tells how many equal parts make the whole.

Adding/Subtracting - find a common denominator first, then add or subtract numerators.

$$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$ $$\frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}$$

Multiplying - multiply straight across (numerator × numerator, denominator × denominator), then simplify.

$$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$$

Dividing - keep the first fraction, flip the second (reciprocal), multiply.

$$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1 \frac{1}{8}$$

Mixed numbers - convert to improper fractions first, then operate.

$2 \frac{1}{3} = \frac{7}{3}$, so $2 \frac{1}{3} \times 1 \frac{1}{2} = \frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = \frac{7}{2} = 3 \frac{1}{2}$

Decimals

Decimals are fractions in base 10. The same operations apply, just watch the decimal point.

Adding/Subtracting - line up the decimal points, pad with zeros if needed.

$$\begin{array}{rrrrr} & 1 & 2 & .3 & 4 \\ {+} & & 5 & .7 & 0 \\ \hline & 1 & 8 & .0 & 4 \end{array}$$

Multiplying - ignore the decimals, multiply as whole numbers, then count total decimal places and place the point.

$1.6 \times 2.5$: multiply $16 \times 25 = 400$, total 2 decimal places → $4.00 = 4$

Dividing - make the divisor a whole number by shifting both decimals the same number of places, then divide normally.

$7.2 \div 0.6$: shift both 1 place → $72 \div 6 = 12$

Percentages

Percent means “per hundred.” $25\% = \frac{25}{100} = 0.25$

Converting: Move the decimal 2 places left to go from percent to decimal ($45\% → 0.45$). Move 2 places right to go from decimal to percent ($0.6 → 60\%$). For fractions, divide then convert ($\frac{3}{8} = 0.375 = 37.5\%$).

Finding a percent of a number: Convert to decimal and multiply. $15\%$ of 80 dollars $= 0.15 \times 80 = 12$ dollars

Percent increase/decrease: Find the change amount, then add or subtract. A 60-dollar item marked up 10%: $0.10 \times 60 = 6$, new price $= 66$ dollars. A 200-dollar item on sale 30% off: $0.30 \times 200 = 60$ off, sale price $= 140$ dollars.

Mixed Practice

Try these problems that combine skills from across the section:

1. You buy 3 items at 12.49 dollars, 8.75 dollars, and 24.99 dollars. What’s the total? $12.49 + 8.75 + 24.99 = 46.23 \text{ dollars}$

2. A recipe calls for $2 \frac{1}{2}$ cups of flour. You want to make $\frac{2}{3}$ of the recipe. How much flour? $\frac{5}{2} \times \frac{2}{3} = \frac{10}{6} = \frac{5}{3} = 1 \frac{2}{3}$ cups

3. A store has 456 items. 35% are on sale. How many on sale? $0.35 \times 456 = 159.6$ → about 160 items

4. Gas is 3.89 dollars per gallon. For 12.5 gallons, total cost? $3.89 \times 12.5 = 48.625$ → 48.63 dollars

5. You have $3 \frac{3}{4}$ gallons of paint and use $1 \frac{2}{3}$ gallons. How much left?

$\begin{aligned} \frac{15}{4} - \frac{5}{3} &= \frac{45}{12} - \frac{20}{12} \\[1em] &= \frac{25}{12} = 2 \frac{1}{12} \text{ gallons} \end{aligned}$

6. A 75-dollar jacket is 20% off, then 8% tax on the sale price. Final cost?

$\begin{aligned} \text{Discount: } 0.20 \times 75 &= 15 \\[1em] \text{Sale price: } 75 - 15 &= 60 \text{ dollars} \\[1em] \text{Tax: } 0.08 \times 60 &= 4.80 \\[1em] \text{Final: } 60 + 4.80 &= 64.80 \text{ dollars} \end{aligned}$

Real-World Applications

Every topic in this section shows up in daily life. Place value helps you read prices and measurements. Whole number operations handle budgets, quantities, and schedules. Fractions come up in cooking, building, and splitting things. Decimals are money, measurements, and stats. Percentages drive discounts, tips, taxes, and interest. Together, these skills let you handle receipts, bills, recipes, and quick estimates without reaching for a calculator.

You’ve Got This

You’ve covered a lot, from basic place value all the way to percent problems, and every skill builds on the last. If something feels rusty, go back and review that lesson. Try working through a real receipt or bill using the skills from each section. You’re much stronger than when you started.

Quiz

Round 456.789 to the nearest tenth.

• A.456.8
• B.456.7
• C.457.0
• D.456.79

What is 3/4 × 2/5?

• A.3/10
• B.5/9
• C.6/9
• D.1/2

Subtract 5/6 - 1/4.

• A.7/12
• B.4/2
• C.1/2
• D.2/3

What is 2.3 × 1.4?

• A.3.22
• B.3.12
• C.3.32
• D.2.22

A 120-dollar item is 25% off. Sale price?

• A.90 dollars
• B.30 dollars
• C.150 dollars
• D.95 dollars

Add 12.75 dollars + 8.99 dollars + 3.50 dollars.

• A.25.24 dollars
• B.24.24 dollars
• C.25.14 dollars
• D.24.14 dollars

Divide 3/4 ÷ 2/3.

• A.9/8
• B.6/12
• C.1/2
• D.2/3

A price drops from 80 dollars to 60 dollars. Percent decrease?

• A.25%
• B.20%
• C.30%
• D.15%

What is $\frac{2}{3} + \frac{1}{4}$?

• A.$\frac{11}{12}$
• B.$\frac{3}{7}$
• C.$\frac{8}{12}$
• D.$\frac{5}{6}$

Round 3.847 to the nearest hundredth.

• A.3.85
• B.3.84
• C.3.80
• D.3.90

What is 456 + 789?

• A.1,245
• B.1,235
• C.1,345
• D.1,145

Subtract 5,003 - 2,478.

• A.2,525
• B.2,535
• C.3,525
• D.2,625

What is 25 × 16?

• A.400
• B.375
• C.425
• D.350

Divide 936 ÷ 12.

• A.78
• B.72
• C.82
• D.76

In the number 7,294, what is the value of the digit 2?

• A.200
• B.20
• C.2
• D.2,000

What is 1.5 × 0.4?

• A.0.60
• B.6.0
• C.0.06
• D.0.54

Convert $\frac{7}{8}$ to a decimal.

• A.0.875
• B.0.78
• C.0.87
• D.0.75

What is 15% of 80?

• A.12
• B.15
• C.8
• D.10

Simplify $\frac{18}{24}$.

• A.$\frac{3}{4}$
• B.$\frac{2}{3}$
• C.$\frac{9}{12}$
• D.$\frac{6}{8}$

A 90-dollar item has 7% sales tax. What is the total cost?

• A.96.30 dollars
• B.97.00 dollars
• C.96.00 dollars
• D.93.70 dollars