Where this sits. This is one of the five number families — the arithmetic that a little under half the test is made of, and that every algebra family is built out of. Every family page on this site has the same eight parts, in the same order, so once you know the shape of one you know the shape of all of them. Where a part is empty, it says so.
DrillsTwenty-seven drills for this family, one skill each, ten problems at a timeGo to the drills ↓Place value — what a digit is worth by where it stands — rounding, comparing, and adding through dividing, including the order of operations: the agreed sequence for doing a calculation so that everyone arrives at the same answer. Factors, multiples, and primes belong here too: which numbers build a number, and which it builds.
On the test, this family rarely appears under its own name. It appears inside every other question: a percent you have to divide, a perimeter you have to add, an equation with a multiplication buried in it. Slow arithmetic costs you time on questions that are not about arithmetic at all, which is why this family comes first.
What the test asks in this family
- Place value — what a digit is worth by where it standsQuiz 1
- Reading and writing large numbersQuiz 1
- Comparing and ordering whole numbersQuiz 2
- RoundingQuiz 2
- Estimating, to check an answer before you trust itQuiz 2
- Adding and subtracting, with carrying and borrowingQuiz 3
- Multiplying multi-digit numbersQuiz 4
- Long division, and what to do with the remainderQuiz 5
- Order of operationsQuiz 6
- Factors, multiples, and primesQuiz 7
- Divisibility testsQuiz 7
- Prime factorizationQuiz 7
- The times tables, by heartTimes Tables
All 13 have something on this site.
Seven questions across this family. Type an answer and press Check. You get the reasoning for every one, right or wrong.
- In 4,208,671, what is the 2 worth? 200,000. A digit is worth what its place says it is worth. Counting from the right, the places go ones, tens, hundreds, then thousands, ten thousands, hundred thousands. The 2 stands in the hundred thousands place, so it is worth two hundred thousand — not two.
- Round 47,850 to the nearest thousand. 48,000. The thousands digit is the 7, so you are choosing between 47,000 and 48,000. Look at the digit to its right: an 8. Five or more rounds up, so the 7 becomes an 8 and everything after it becomes zero.
- 8,003 − 1,476 = 6,527. This is the hard kind of subtraction, because you have to borrow across two zeros. You cannot take 6 from 3, and the neighbor is a 0, and its neighbor is a 0 too. Go left until you find something to borrow from — the 8 — and trade down through the zeros: 8,003 becomes 7 thousands, 9 hundreds, 9 tens, and 13 ones.
- 406 × 7 = 2,842. The zero in the middle is where people go wrong. Work right to left: 7 × 6 is 42, write the 2 and carry the 4; 7 × 0 is 0, plus the carried 4 is 4; 7 × 4 is 28. The zero still has to be multiplied — it just contributes nothing except a place to put the carry.
- 7 + 3 × 4 = 19, not 40. Multiplication happens before addition, whatever order the numbers are written in. So this is 7 + 12. The order of operations is not a rule about what is more important; it is an agreement, so that everyone doing this calculation arrives at the same number.
- A carton holds 8 jars. You have 253 jars. How many full cartons can you fill? 31. This is 253 ÷ 8, which is 31 with 5 left over. The question asks for full cartons, so the 5 left over do not make a carton and the answer is 31. Every division question with a remainder asks you to decide what to do with it: here you drop it, but a question about how many cartons you need to hold 253 jars would want 32.
- Is 51 a prime number? No. It looks prime, which is why it is worth asking. 51 = 3 × 17, so it has factors other than 1 and itself. The quick test: add its digits, 5 + 1 = 6, and 6 is divisible by 3, so 51 is divisible by 3.
Missed one? The seven quizzes of the Seahorse Series are in Learn it just below, and Times Tables for Grown-Ups under Work it by hand drills the multiplication facts that make all of this faster.
Want more than these seven questions? Flyover I asks about all fourteen families in one sitting and ends in a grid that says where to start; Flyover II asks them again later, with different questions, so you can see what has moved.
- Quiz 1Place ValueWhat each digit is worth, and why.
- Quiz 2Comparing, Ordering, and RoundingWhich is bigger, and to the nearest what.
- Quiz 3Addition and Subtraction with RegroupingCarrying and borrowing, done plainly.
- Quiz 4Multi-Digit MultiplicationThe long way, set out so it works every time.
- Quiz 5DivisionLong division, and what a remainder means.
- Quiz 6Order of OperationsThe agreed sequence, so everyone gets the same answer.
- Quiz 7Factors, Multiples, and PrimesWhich numbers build a number, and which it builds.
These seven are the Seahorse Series, the first leg of the quiz series. Each one teaches its topic first, with a guide and three worked examples, then asks ten questions and explains every answer.
Twenty-seven short drills, one skill each, in order. Each starts with two worked examples, then gives you ten problems (or five) and the reasoning for every answer. Each one prints, with the answer key on the last page. When you get all ten right, or all five right twice in a row, the drill gets a check mark on the Drills page.
- 1.01Place value: what a digit is worth
- 1.02Write numbers in words and in digits
- 1.03Compare and order whole numbers
- 1.04Round to the nearest ten, hundred, thousand
- 1.05Addition facts to 20
- 1.06Subtraction facts to 20
- 1.07Add multi-digit numbers, carrying
- 1.08Subtract multi-digit numbers, borrowing (including across zeros)
- 1.09Times tables: 2, 3, 4, 5
- 1.10Times tables: 6, 7, 8, 9
- 1.11Times tables: 10, 11, 12
- 1.12Division facts: the times tables backward
- 1.13Multiply by a one-digit number
- 1.14Multiply two- and three-digit numbers
- 1.15Multiply and divide by 10, 100 and 1,000
- 1.16Divide by a one-digit number, with remainders
- 1.17Long division by a two-digit number
- 1.18Estimate an answer by rounding first
- 1.19Order of operations: two operations
- 1.20Order of operations: parentheses and exponents
- 1.21List all the factors of a number
- 1.22Multiples, and the least common multiple
- 1.23The greatest common factor
- 1.24Prime or composite; prime factorization
- 1.25Divisibility by 2, 3, 5, 9 and 10
- 1.26Word problems: which operation? (one step) · calculator allowed
- 1.27Word problems: two steps · calculator allowed
- Times Tables for Grown-Ups · the twelve tables, each one in its own material, with a fold-it-in-half view that cuts the work by half, and practice that leans toward the facts you miss.
- Times Tables for Grown-Ups · the same page, made to print.
- Khan Academy, math · free videos and practice. Search it for place value, long division, order of operations. These are the clips we start from in class.
- Math Antics · short, plain videos on the basics.
- Light & Salt Learning · free GED classes online, with playlists by topic.
- Get Sum Math · GED math on video, one topic at a time.
- Kaplan GED Test Prep and the McGraw-Hill Mathematical Reasoning Workbook · look up whole numbers and the four operations in the contents of either.
Straight on to signed numbers — positives and negatives. The four operations you have just practiced are the same four operations, and the only new thing is what happens when the numbers drop below zero. The test uses signed numbers constantly once algebra starts, because half the numbers in algebra are negative.