Integers, decimals, and fractions are foundational number types used throughout mathematics. On the Praxis Core exam, being able to interpret and operate on these forms helps you handle multi-step problems accurately. This chapter builds a clear understanding of each number type and shows how to use them in practical situations.

Integers

Integers include zero, positive whole numbers, and their opposites (negative whole numbers). You’ll often use integers to describe direction and change - for example, gains and losses, temperature changes, or positions above and below a reference point.

On a number line:

• Positive integers lie to the right of zero.
• Negative integers lie to the left of zero.
• Zero is the neutral midpoint.

The sign of an integer affects how operations work. Addition and subtraction depend on both direction and magnitude, while multiplication and division follow consistent sign rules. Once you recognize these patterns, you can rely less on memorization and more on reasoning.

Adding and subtracting integers

You can think of adding integers as combining movements on the number line.

• If both integers have the same sign, add their magnitudes and keep the sign.
• If the integers have different signs, subtract the smaller magnitude from the larger magnitude. The result takes the sign of the number with the larger absolute value.

Subtraction becomes easier if you rewrite it as “add the opposite.” For example, rewriting $7-(-3)$ as $7+3$ makes the operation and the number-line meaning clearer.

Example: Add integers Find the value of $5+(-3)$.

• Different signs
• Subtract magnitudes: $5-3=2$
• Larger absolute value is positive

Answer: $2$

Example: Add integers Find the value of $-7+12$.

(spoiler)

• Different signs
• $12-7=5$
• Larger absolute value is positive

Answer: $5$

Example: Subtract integers Find the value of $7-(-3)$.

• Rewrite: $7+3$
• Combine to get $10$

Answer: $10$

Example: Subtract integers Find the value of $-9-(-4)$.

(spoiler)

• Rewrite: $-9+4$
• Subtract magnitudes → $9-4=5$
• Larger absolute value is negative

Answer: $-5$

Multiplying and dividing integers

Multiplying and dividing integers follow the same sign patterns:

• If the signs are the same, the result is positive.
• If the signs are different, the result is negative.

To compute the magnitude, multiply or divide the absolute values, then apply the sign rule.

Important reminders:

• Same signs → positive result
• Different signs → negative result
• Multiply or divide magnitudes normally

Example: Multiply integers Find the value of $-6\times -5$.

• Same signs → positive
• $6\times 5=30$

Answer: $30$

Example: Multiply integers Find the value of $-8\times 3$.

(spoiler)

• Different signs → negative
• $8\times 3=24$

Answer: $-24$

Decimals

Decimals are numbers written in base ten. You’ll see them often in measurements, money, and scientific values. Each digit to the right of the decimal point represents a smaller place value (tenths, hundredths, thousandths, and so on). Place value is what lets you compare decimals and compute with them correctly.

Decimal operations follow the same basic rules as whole-number operations, but you need to manage the decimal point carefully:

• When adding or subtracting, line up decimal points.
• When multiplying, multiply first, then place the decimal based on the total number of decimal places.
• When dividing, shift the decimal in the divisor to make it a whole number (and shift the dividend the same amount).

Key ideas:

• Align decimal points for addition and subtraction
• Place value determines size
• Multiplication depends on counting decimal places
• Division requires shifting the decimal in the divisor

Adding and subtracting decimals

To add or subtract decimals, align the decimal points so each digit stays in its correct place value. If needed, you can add zeros to the right of a decimal to make the numbers line up.

Regrouping (borrowing and carrying) works the same way as it does with whole numbers. The key is that each place is worth ten times the place to its right, and that relationship continues across the decimal point. After you compute, bring the decimal point straight down into the answer.

Example: Add decimals Find the value of $3.25+4.75$.

$$>\begin{array}{r}3.25\\ +4.75\\ 8.00\end{array}$$

• Add hundredths and tenths with carrying
• Add ones normally

Answer: $8$

Example: Add decimals Find the value of $3.28+4.63$.

(spoiler)

• $8+3=11$
• $2+6+1=9$
• $3+4=7$

$$>\begin{array}{r}3.28\\ +4.63\\ 7.91\end{array}$$

Answer: $7.91$

Example: Subtract decimals Find the value of $5.40-2.18$.

$$>\begin{array}{r}5.40\\ -2.18\\ 3.22\end{array}$$

• Borrow when subtracting tenths and hundredths

Answer: $3.22$

Example: Subtract decimals Find the value of $7.05-3.80$.

(spoiler)

• Borrow from the ones place
• Subtract tenths and hundredths

Answer: $3.25$

Multiplying decimals

To multiply decimals, start by ignoring the decimal points and multiplying as if the numbers were whole numbers. Then place the decimal point in the product using this rule:

• Count the total number of digits to the right of the decimal in both factors.
• The product must have that same total number of decimal places.

This works because moving a decimal point is the same as multiplying by a power of ten. You temporarily remove the decimals to simplify the multiplication, then put the decimal back so the final value has the correct magnitude.

Example: Multiply decimals Find the value of $0.5\times 0.2$.

• Multiply $5\times 2=10$
• Two total decimal places

Answer: $0.1$

Example: Multiply decimals Find the value of $3.6\times 0.04$.

(spoiler)

• Multiply $36\times 4=144$
• Three total decimal places

Answer: $0.144$

Example: Multiply decimals Find the value of $2.45\times 8.43$.

(spoiler)

• Multiply $245\times 843=206535$
• Four decimal places

Answer: $20.6535$

Dividing decimals

To divide decimals, first make the divisor a whole number.

• Shift the decimal point in the divisor to the right until it becomes a whole number.
• Shift the decimal point in the dividend the same number of places.
• Divide as you would with whole numbers.

This works because multiplying both the dividend and divisor by the same power of ten doesn’t change the quotient.

Example: Divide decimals Find the value of $2.4÷0.6$.

• Shift one place right → $24÷6$
• Compute normally

Answer: $4$

Example: Divide decimals Find the value of $4.8÷0.2$.

(spoiler)

• Shift one place → $48÷2$
• Compute normally

Answer: $24$

Fractions

Fractions represent parts of a whole or relationships between quantities. The numerator tells how many parts you have, and the denominator tells how many equal parts make up one whole. Fractions show up often in measurement, ratios, and proportional reasoning.

To work smoothly with fractions, you’ll want to be comfortable with operations, comparisons, and conversions between forms.

Helpful reminders:

• Fractions with the same denominator can be added or subtracted directly
• Using the least common denominator simplifies calculations
• Multiplication and division do not require matching denominators
• Converting between mixed and improper forms often simplifies work

Adding and subtracting fractions

You can only add or subtract fractions directly when they have a common denominator.

• If the denominators are the same, add or subtract the numerators and keep the denominator.
• If the denominators are different, rewrite the fractions using equivalent forms with the least common denominator (LCD). Then add or subtract the numerators.

Afterward, simplify the result when possible. If the numerator is larger than the denominator, you can rewrite the answer as a mixed number.

Example: Add fractions Find the value of $\frac{1}{5}+\frac{2}{5}$.

• Same denominator
• $1+2=3$

Answer: $=\frac{3}{5}$

Example: Add fractions Find the value of $\frac{1}{4}+\frac{1}{6}$.

• LCD = $12$
• Rewrite as $\frac{3}{12}+\frac{2}{12}$
• Combine to get $\frac{5}{12}$

Answer: $\frac{5}{12}$

Example: Add fractions Find the value of $\frac{3}{4}+\frac{2}{5}$.

(spoiler)

• LCD = $20$
• Convert to $\frac{15}{20}$ and $\frac{8}{20}$
• Combine → $\frac{23}{20}$
• Mixed number → $1\frac{3}{20}$

Answer: $\frac{23}{20}$

Multiplying fractions

To multiply fractions:

• Multiply the numerators.
• Multiply the denominators.
• Simplify if possible.

You don’t need a common denominator.

It can also help to think of multiplication as scaling:

• Multiplying by a fraction less than 1 makes the value smaller.
• Multiplying by a number greater than 1 makes the value larger.

Example: Multiply fractions Find the value of $\frac{2}{3}\times \frac{4}{5}$.

• Multiply numerators and denominators

Answer: $\frac{8}{15}$

Dividing fractions

To divide fractions, use the Keep-Change-Flip method:

• Keep the first fraction.
• Change division to multiplication.
• Flip the second fraction to its reciprocal.

Then multiply.

This works because dividing by a fraction is the same as multiplying by its reciprocal.

Example: Divide fractions Find the value of $\frac{3}{4}÷\frac{2}{5}$.

• Rewrite as $\frac{3}{4}\times \frac{5}{2}$
• Multiply across

Answer: $\frac{15}{8}$

Example: Divide fractions Find the value of $\frac{5}{8}÷\frac{2}{3}$.

(spoiler)

• Rewrite as $\frac{5}{8}\times \frac{3}{2}$
• Multiply across

Answer: $\frac{15}{16}$

Mixed numbers and improper fractions

Mixed numbers and improper fractions are two equivalent ways to represent values greater than one.

• A mixed number combines a whole number and a fraction.
• An improper fraction has a numerator greater than or equal to its denominator.

Mixed numbers are often easier to read, while improper fractions are usually easier to compute with. Because of that, it’s common to convert mixed numbers to improper fractions before doing operations, then convert back at the end if needed.

Converting between forms

To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator.
• Keep the denominator.

Example: Convert mixed to improper Convert $3\frac{2}{5}$.

• $3\times 5+2=17$

Answer: $\frac{17}{5}$

To convert an improper fraction to a mixed number:

• Divide the numerator by the denominator.
• The quotient is the whole number.
• The remainder becomes the new numerator (over the same denominator).

Example: Convert improper to mixed Convert $\frac{19}{6}$.

• $19÷6=3$ remainder $1$

Answer: $3\frac{1}{6}$

Adding mixed numbers

To add mixed numbers, a reliable approach is:

• Convert each mixed number to an improper fraction.
• Add the fractions using a common denominator.
• Convert the result back to a mixed number if desired.

Example: Add mixed numbers Find the value of $2\frac{1}{4}+1\frac{2}{3}$.

• Convert to $\frac{9}{4}$ and $\frac{5}{3}$
• LCD $=12$ → $\frac{27}{12}+\frac{20}{12}$
• Sum $=\frac{47}{12}$

Answer: $3\frac{11}{12}$