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: Subtract integers

Find the value of $7-(-3)$.

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

Answer: $10$

Watch out: When a subtraction problem has double negatives, like $-9-(-4)$, be careful to rewrite it as addition before combining: $-9+4=-5$. A common mistake is dropping the sign of the second number and computing $-9-4=-13$ instead of the correct $-5$.

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

Example: Multiply integers

Find the value of $(-6)\times (-5)$.

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

Answer: $30$

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.

Adding and subtracting decimals

To add or subtract decimals, align the decimal points so each digit stays in its correct place value. If needed, 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. Each place is worth ten times the place to its right, and that relationship continues across the decimal point. After computing, bring the decimal point straight down into the answer.

Example: Add decimals

Find the value of $3.28+4.63$.

• Hundredths: $8+3=11$; write $1$, carry $1$
• Tenths: $2+6+1=9$; write $9$
• Ones: $3+4=7$

Answer: $7.91$

Example: Subtract decimals

Find the value of $5.40-2.18$.

• Borrow 1 tenth: tenths digit goes from $4$ to $3$, hundredths becomes $10$
• Hundredths: $10-8=2$
• Tenths: $3-1=2$
• Ones: $5-2=3$

Answer: $3.22$

Comparing decimals

To compare decimals, look at each place value from left to right - the first place where the digits differ tells you which number is larger. A useful technique is to add trailing zeros so both numbers have the same number of decimal places, then compare as if they were whole numbers.

Example: Compare decimals

Which is greater: $0.45$ or $0.4$?

• Rewrite $0.4$ as $0.40$ so both have two decimal places.
• Compare: $0.45$ vs $0.40$
• In the hundredths place, $5>0$, so $0.45>0.40$.

Answer: $0.45$ is greater.

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 $0.5\times 0.2$.

• $0.5$ has 1 decimal place; $0.2$ has 1 decimal place → 2 total decimal places
• Multiply whole numbers: $5\times 2=10$
• Place the decimal 2 places from the right: $10\to 0.10$
• Drop the trailing zero: $0.10=0.1$

Answer: $0.1$

Watch out: Count the decimal places in both factors, not just one. For example, $0.04\times 0.3$ has 3 total decimal places ($2+1$), so $4\times 3=12$ becomes $0.012$, not $0.12$. Always count from the right of the product.

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 to the right.
• 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: $4$

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.

Comparing fractions

To compare fractions, convert them to a common denominator (or convert both to decimals) and compare the resulting values.

Example: Compare fractions

Which is greater: $\frac{3}{5}$ or $\frac{5}{8}$?

• LCD = $40$: rewrite as $\frac{24}{40}$ and $\frac{25}{40}$.
• $25>24$, so $\frac{5}{8}>\frac{3}{5}$.

Answer: $\frac{5}{8}$ is greater.

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 $\frac{1}{4}+\frac{1}{6}$.

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

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

Example: Subtract fractions with unlike denominators

Find the value of $\frac{5}{6}-\frac{1}{4}$.

• LCD = $12$
• Rewrite as $\frac{10}{12}-\frac{3}{12}$
• Subtract numerators: $10-3=7$

Answer: $\frac{7}{12}$

Watch out: You only need a common denominator when adding or subtracting fractions. When multiplying or dividing fractions, skip finding the LCD - just multiply (or use Keep-Change-Flip) directly.

Multiplying fractions

To multiply fractions:

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

You don’t need a common denominator.

It also helps 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 $\frac{4}{9}\times \frac{3}{8}$.

• Before multiplying, look for common factors across numerators and denominators.
• $4$ and $8$ share a factor of $4$: $\frac{4}{8}\to \frac{1}{2}$
• $3$ and $9$ share a factor of $3$: $\frac{3}{9}\to \frac{1}{3}$
• Rewrite: $\frac{1}{3}\times \frac{1}{2}$
• Multiply: $\frac{1\times 1}{3\times 2}=\frac{1}{6}$

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

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.

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: $\frac{15}{8}$

Mixed numbers and improper fractions

Now that you’re comfortable with fraction operations, it’s worth extending those skills to values greater than one - which brings us to 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: $3\frac{1}{6}$

Operations with mixed numbers

The convert-to-improper strategy works for all four operations: convert first, operate on the improper fractions, then convert back if needed.

Example: Add mixed numbers

Find the value of $1\frac{1}{2}+2\frac{1}{3}$.

• Convert: $1\frac{1}{2}=\frac{3}{2}$ and $2\frac{1}{3}=\frac{7}{3}$
• LCD = $6$: rewrite as $\frac{9}{6}+\frac{14}{6}$
• Add: $\frac{23}{6}$
• Convert back: $23÷6=3$ remainder $5$

Answer: $3\frac{5}{6}$

Converting between fractions and decimals

Fractions and decimals are two ways of writing the same value, and converting between them is a key skill for the Praxis Core exam.

Fraction to decimal: Divide the numerator by the denominator.

Example: Convert a fraction to a decimal

Convert $\frac{3}{4}$ to a decimal.

• Divide: $3÷4=0.75$

Answer: $0.75$

Decimal to fraction: Write the decimal as a fraction using place value, then simplify.

Example: Convert a decimal to a fraction

Convert $0.6$ to a fraction.

• $0.6$ is six tenths → $\frac{6}{10}$
• Simplify: $\frac{6}{10}=\frac{3}{5}$

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