Praxis Core Math 5733 • Content Category I • Chapter 1

Number and Quantity

This is the largest category on the test: about 20 of the 56 questions (36%). None of it is new to you. The goal of this chapter is to make every idea visual so the rules become obvious instead of memorized, and to practice them at the difficulty the exam actually uses: real contexts, multi-step setups, and messy numbers. Work through each diagram, then the worked examples beside it.

By the end of this chapter you will be able to

Solve problems involving integers, decimals, and fractions.
Solve problems involving ratios and proportions.
Solve problems involving percent.
Solve problems involving constant rates (miles per hour, gallons per mile, cubic feet per minute).
Demonstrate an understanding of place value, naming of decimal numbers, and ordering of numbers.
Demonstrate an understanding of the properties of whole numbers (factors, multiples, even and odd, prime, divisibility).
Identify counterexamples to statements using basic arithmetic.
Solve real-life problems by identifying relevant numbers, information, or operations, including rounding.
Solve problems involving units, including unit conversion and measurement.

A. Integers, Decimals, and Fractions

What an integer is

An integer is a whole number or its negative, with no fraction or decimal part: …, -3, -2, -1, 0, 1, 2, 3, …

So -16, 0, and 198 are integers, but 12, 1.1, and 3.5 are not. Integers live evenly spaced on the number line, with zero in the middle and the negatives mirroring the positives.

The integer number line

A real problem that uses an integer

Temperature change

At dawn the temperature is -6°F. By early afternoon it has risen 19 degrees. What is the new temperature? Adding moves you to the right on the line.

-6 + 19 = 13. The new temperature is 13°F.

Comparing integers

On the number line, the number farther to the right is always greater. For negatives this feels backwards: the one closer to zero is larger.

Example 1

17 > -53

Any positive number beats any negative number.

Example 2

-42 < -39

-42 sits farther left than -39, so it is smaller.

Common trap: treating a "bigger looking" negative as larger. -42 looks bigger than -39, but it is less. Think temperature: -42° is colder than -39°.

Three number sets worth knowing

Name	The numbers	Examples
Counting numbers	{ 1, 2, 3, 4, … }	7, 18, 2,061
Whole numbers	{ 0, 1, 2, 3, … }	0, 27, 398
Integers	{ …, -2, -1, 0, 1, 2, … }	-15, 0, 1,102

Each set sits inside the next: every counting number is whole, every whole number is an integer.

Adding and subtracting integers

Same signs: add the values and keep the sign. Different signs: subtract the smaller value from the larger and take the sign of the larger. Subtracting a number is the same as adding its opposite.

-8 + 3 = -5

-6 + (-9) = -15

5 - 12 = -7

-4 - (-11) = 7

23 - (-7) = 30

(-14) - 1 = -15

Look at -4 - (-11): subtracting -11 flips to -4 + 11 = 7. Two minus signs side by side become a plus.

Multiplying and dividing integers

The sign rule is the whole game

× or ÷	positive	negative
positive	+	-
negative	-	+

Same signs → positive. Different signs → negative.

(-7) × 8 = -56

(-6) × (-9) = 54

11 × (-2) = -22

-72 ÷ 9 = -8

-54 ÷ (-6) = 9

Common trap: two negatives multiplied give a positive. (-6) × (-9) = +54, not -54.

Decimal operations

Add and subtract: line up the points

14.70

+ 3.85

18.55

Stack the decimal points, pad with a zero so the columns match, then add as usual. 20.4 - 6.73 works the same way and gives 13.67.

Multiply: count the decimal places

1.2 × 0.05 = 0.06

Multiply as if there were no points: 12 × 5 = 60. The factors have 1 + 2 = 3 decimal places, so the answer needs 3: 0.060 = 0.06.

Divide: shift the point

4.2 ÷ 0.6 = 7

Move both points right one place to make the divisor a whole number: 42 ÷ 6 = 7.

Fraction operations

Add: rebuild in a common denominator

Thirds and quarters both fit into twelfths. Rewrite, then add the numerators.

Subtract: same common size, take away

9 twelfths minus 4 twelfths leaves 5 twelfths.

Multiply: straight across (area model 2/3 × 3/4)

2 of 3 rows by 3 of 4 columns shades 6 of 12 cells. Multiply tops and bottoms: 2×3 / 3×4 = 6/12 = 1/2.

Divide: keep, change, flip the divisor

Dividing by 2 is the same as multiplying by 1/2.

On the exam: a single item may mix forms, for example asking you to add a fraction to a decimal. Convert to one form first (often decimals), keep negatives straight, and watch which operation the wording calls for.

B. Ratios and Proportions

A ratio compares two amounts

A ratio tells you how much of one thing there is compared to another. It can be written with a colon (5 : 2) or with the word "to" (5 to 2). Both mean 5 of the first for every 2 of the second.

A muffin recipe uses 5 cups flour to 2 cups sugar

flour to sugar is 5 to 2.

Equivalent ratios: scale both parts the same way

Triple the recipe and the ratio holds

15 : 6 is the same ratio as 5 : 2 because both parts were multiplied by 3. Dividing both parts by a common number (reducing) also keeps the ratio.

Part-to-part versus part-to-whole

A reading group has 12 women and 8 men (20 people total)

Part to part

women : men = 12 : 8 = 3 : 2

Part to whole

women : all = 12 : 20 = 3 : 5 (also 35)

Common trap: read the wording carefully. "Women to men" (part to part) is different from "women out of the whole group" (part to whole). Mixing them is the most common ratio mistake.

A proportion sets two ratios equal

2/5 and 6/15 fill the same fraction of the bar

Two ratios that reduce to the same value are in proportion. 6/15 reduces to 2/5.

Solving a proportion: cross-multiply

A recipe for 4 servings uses 6 eggs. How many eggs for 10 servings?

Cross-multiply the two known diagonals, then divide by the remaining number. Keep matching units across from each other (eggs over servings on both sides).

Proportions in the real world: similar triangles

How tall is the flagpole?

At the same moment, a 3 ft yardstick casts a 2 ft shadow and a flagpole casts a 24 ft shadow. The sun makes the same angle for both, so the triangles are similar and their height-to-shadow ratios match.

The flagpole is 36 feet tall.

A three-part ratio

Trail mix is mixed peanuts : raisins : chocolate = 4 : 3 : 2. You have 12 oz of chocolate. How much of each?

	Peanuts	Raisins	Chocolate
Ratio	4	3	2
You have	24 oz	18 oz	12 oz

Chocolate went from 2 to 12, a scale factor of ×6. Multiply every part by 6: 4×6 = 24, 3×6 = 18. The ratio 24 : 18 : 12 reduces right back to 4 : 3 : 2.

C. Percent

Percent means "per 100"

37 shaded out of 100

A percent is just a fraction with a denominator of 100. 1% means 1 out of 100, 100% means the whole thing, and 200% means twice as much.

One value, three forms

Percent	Decimal	Fraction (reduced)
50%	0.5	1/2
25%	0.25	1/4
75%	0.75	3/4
20%	0.20	1/5
12.5%	0.125	1/8

Percent to decimal: divide by 100 (move the point two places left). Decimal to percent: multiply by 100 (move it two places right). Percent to fraction: put it over 100 and reduce.

Finding a percent of a number

An auditorium seats 1,250. On opening night 84% of seats are filled.

part = percent × whole. 0.84 × 1,250 = 1,050 seats filled.

Finding what percent one number is of another

A teacher has graded 45 of her 60 essays.

4560 = 0.75 = 75%

part ÷ whole, then multiply by 100. She is 75% finished.

Percent increase and decrease

Increase: a salary rises from $48,000 to $51,600

$48,000 + $3,600 raise

change ÷ original = 3,600 ÷ 48,000 = 0.075 = 7.5% increase.

Decrease: a $140 jacket is 35% off

0.35 × $140 = $49 off
$140 - $49 = $91

Find the discount, then subtract it from the original. Sale price = $91.

Common trap: for percent change, always divide by the original amount, not the new one. Dividing 3,600 by 51,600 gives about 7%, the classic wrong answer.

D. Constant Rates

A rate compares two quantities measured in different units, usually written as "this per that." A unit rate reduces the second quantity to 1 by dividing.

Unit rate: 180 miles driven in 3 hours

180 miles3 hours = 60 miles per hour

Divide to land on "per 1 hour." Distance, rate, and time are tied together by distance = rate × time.

The three rates named on the exam

Miles per hour

A train covers 330 miles in 5.5 hours.

330 ÷ 5.5 =

60 mph

Gallons per mile

A delivery truck burns 8 gallons over 100 miles.

8 ÷ 100 =

0.08 gal/mi

Cubic feet per minute

A fan moves 1,200 cubic feet of air in 8 minutes.

1,200 ÷ 8 =

150 cfm

On the exam: rate items often ask you to extend a constant rate. At 0.08 gallons per mile, a 250-mile trip needs 0.08 × 250 = 20 gallons. Set up the unit rate first, then multiply.

E. Place Value, Naming Decimals, and Ordering

Every seat is worth ten times the seat on its right

The whole-number board for 3,408,150

MILLIONS		HUNDRED THOUS	TEN THOUS	THOUS		HUND REDS	TENS	ONES
3	,	4	0	8	,	1	5	0

← each seat ×10 going left each seat ÷10 going right →

Read it: three million, four hundred eight thousand, one hundred fifty. The 4 alone is worth 400,000.

To the right of the decimal point: 6.274

ONES		TENTHS	HUND REDTHS	THOUS ANDTHS
6	.	2	7	4

Tenths, hundredths, thousandths each shrink by ten. The 7 is worth 7 hundredths.

Naming a decimal in words

Number	In words
0.6	six tenths
0.27	twenty-seven hundredths
0.305	three hundred five thousandths
24.07	twenty-four and seven hundredths

The decimal point is read as the word "and." The last digit's place name (tenths, hundredths, thousandths) finishes the name.

Ordering numbers: pad with zeros, then read left to right

Order 2.5, 2.45, 2.405, 2.54 from least to greatest

2 . 5 0 0

2 . 4 5 0

2 . 4 0 5

2 . 5 4 0

Give every number the same number of decimal places by adding zeros, then compare seat by seat. Order: 2.405 < 2.45 < 2.5 < 2.54.

Common trap: "more digits means bigger." False for decimals. 2.405 has the most digits but is the smallest.

F. Properties of Whole Numbers

Factors: numbers that multiply to give a number

Factors come in pairs

Every factor pair of 36: 1×36, 2×18, 3×12, 4×9, 6×6. So the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Multiples: the times table, going on forever

Multiples of 7

Factors divide into a number (a short, finite list). Multiples are what you get by multiplying a number by 1, 2, 3, … (an endless list).

Even and odd

A number is even if its ones digit is 0, 2, 4, 6, or 8 (divisible by 2). Otherwise it is odd.

3,486 is even · 52,071 is odd.

Prime and composite

A prime has exactly two factors, 1 and itself. A composite has more.

Primes under 20: 2, 3, 5, 7, 11, 13, 17, 19. (2 is the only even prime; 1 is neither.)

Prime factorization of 84 (a factor tree)

Keep splitting until only primes remain: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.

Divisibility rules (no long division needed)

Divisible by	Test	Example
2	ones digit is even	3,486 ✓
3	digit sum is divisible by 3	534 (5+3+4=12) ✓
4	last two digits form a number divisible by 4	1,316 (16) ✓
5	ones digit is 0 or 5	2,045 ✓
6	passes both the 2 test and the 3 test	534 ✓
9	digit sum is divisible by 9	6,381 (6+3+8+1=18) ✓
10	ones digit is 0	7,890 ✓

G. Counterexamples

A counterexample is a single case that makes a general statement false. To disprove a claim that says "always," "every," or "all," you only need to find one example where it breaks down. One counterexample is enough; no further proof is needed.

Claim
"All odd numbers are prime."

Counterexample
9 is odd, but 9 = 3 × 3, so it is not prime.

Claim
"Subtraction is commutative: a - b = b - a."

Counterexample
8 - 3 = 5, but 3 - 8 = -5. Not equal.

Claim
"Every multiple of 3 is a multiple of 6."

Counterexample
9 is a multiple of 3, but 9 ÷ 6 is not a whole number.

Claim
"Dividing a number always makes it smaller."

Counterexample
8 ÷ 0.5 = 16. Dividing by a number less than 1 makes it larger.

On the exam: when a question asks which value "disproves" or "is a counterexample to" a statement, test the answer choices one at a time. The correct choice is the one that satisfies the claim's setup but breaks its conclusion.

H. Real-Life Problems and Rounding

Rounding: look at the next digit

Find the place you are keeping, then look at the digit just to its right. If it is 5 or more, round up; if it is 4 or less, round down (keep the digit the same). For whole numbers, replace the dropped digits with zeros.

Round	Next digit	Result
4,728 to the nearest hundred	2	4,700
12,690 to the nearest thousand	6	13,000
3.1416 to the hundredths	1	3.14
1.2735 to the tenths	7	1.3

Pull out only the numbers and operations you need

A word problem with extra information

Ms. Rivera drives 3 miles to the supply store and buys 6 packs of markers at $4.79 each and one $12 box of paper. She pays with a $50 bill. How much change does she get back?

Detail	Use it?
drives 3 miles	No, distance does not affect the cost
6 packs × $4.79	Yes → $28.74
$12 box of paper	Yes → add it
pays with $50	Yes → subtract the total

Total: 28.74 + 12 = $40.74. Change: 50 - 40.74 = $9.26.

Estimate to check: $4.79 is about $5, so 6 × 5 = $30, plus $12 is about $42, leaving roughly $8. The exact answer $9.26 is close, so it is reasonable.

I. Units, Unit Conversion, and Measurement

You are expected to know the everyday U.S. customary and metric units. Conversions are done by multiplying by a conversion factor (a fraction equal to 1) arranged so the unit you do not want cancels out.

U.S. customary

Length	Weight	Capacity
12 in = 1 ft	16 oz = 1 lb	8 fl oz = 1 cup
3 ft = 1 yd	2,000 lb = 1 ton	2 cups = 1 pint
5,280 ft = 1 mile		2 pints = 1 quart
		4 quarts = 1 gallon

Metric (powers of 10)

Length	Mass	Capacity
1 km = 1,000 m	1 kg = 1,000 g	1 L = 1,000 mL
1 m = 100 cm	1 g = 1,000 mg
1 cm = 10 mm

kilo = 1,000 · centi = one hundredth · milli = one thousandth

Convert by canceling units

2.5 miles to feet

2.5 mi × 5,280 ft1 mi = 13,200 ft

"mi" on top cancels "mi" on the bottom, leaving feet. Set the factor so the unwanted unit is in the denominator.

Two quick metric and customary conversions

3,400 g ÷ 1,000 = 3.4 kg

5 gallons × 4 = 20 quarts

A two-step rate conversion: 60 mph to m/s

60 mi1 h × 1,609 m1 mi × 1 h3,600 s

= about 26.8 m/s

"mi" and "h" both cancel, leaving meters per second.

On the exam: when a conversion needs more than one step, chain the factors and cancel units as you go. If the leftover unit matches what the question asks for, your setup is right before you ever multiply.

Quick Reference Card — Chapter 1

Integers: on the number line, farther right = greater. Same signs add and keep the sign; different signs subtract; subtracting flips to adding the opposite.
Signs × ÷: same signs → positive, different signs → negative.
Fractions: common denominator to add or subtract · multiply straight across · divide by keep, change, flip the divisor.
Ratios & proportions: scale both parts by the same number · part-to-part ≠ part-to-whole · cross-multiply to solve a proportion.
Percent: per 100 · part = percent × whole · percent change = change ÷ original.
Rates: divide to get the unit rate (per 1), then multiply · miles per hour, gallons per mile, cubic feet per minute.
Place value & ordering: each seat is 10× the one to its right · pad decimals with zeros before comparing · more digits ≠ larger.
Whole numbers & conversions: factors divide in, multiples come out · primes have exactly two factors · round by the next digit (5 or more rounds up) · convert by canceling units.

What You'll Learn

Free Study Guide - Lesson 1