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SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic Syllabus

The Quantitative Reasoning: Number Sense and Arithmetic syllabus for SSAT (Secondary School Admission Test) in full: 4 chapters, 19 topics and 37 sub-topics, every one listed below, plus 50 Quantitative Reasoning: Number Sense and Arithmetic flashcards with answers.

4Chapters
19Topics
37Sub-topics
50Flashcards

Quantitative Reasoning: Number Sense and Arithmetic syllabus: all 4 chapters

Expand any chapter to see its topics and sub-topics.

  1. Integers and the Number System

     5 topics
    • Place value and number representation
      • Reading and writing whole numbers and decimals
      • Expanded and standard form
      • Rounding and estimation strategies
    • Operations with signed integers
      • Adding and subtracting negatives on a number line
      • Multiplying and dividing signed numbers
      • Absolute value and distance
    • Order of operations (PEMDAS)
      • Nested grouping symbols
      • Common error patterns and traps
    • Properties of operations
      • Commutative and associative properties
      • Distributive property
      • Identity and inverse elements
    • Factors, multiples, and primes
      • Prime factorization
      • Greatest common factor (GCF)
      • Least common multiple (LCM)
      • Divisibility rules

  2. Fractions, Decimals, and Percents

     5 topics
    • Equivalent fractions and simplification
      • Building and reducing fractions
      • Comparing and ordering fractions
    • Operations with fractions
      • Adding and subtracting with unlike denominators
      • Multiplying and dividing fractions
      • Mixed numbers and improper fractions
    • Decimal operations
      • Placement of the decimal point in multiplication and division
      • Terminating versus repeating decimals
    • Converting among fractions, decimals, and percents
      • Benchmark conversions to memorize
    • Percent applications
      • Percent of a number
      • Percent increase and decrease
      • Finding the whole from a part

  3. Ratios, Proportions, and Rates

     5 topics
    • Writing and simplifying ratios
      • Part-to-part versus part-to-whole
      • Three-term ratios
    • Setting up and solving proportions
      • Cross multiplication
      • Scaling recipes and quantities
    • Unit rates and unit pricing
      • Comparing best buys
    • Speed, distance, and time problems
    • Scale drawings and maps

  4. Powers, Roots, and Scientific Notation

     4 topics
    • Exponent rules
      • Product and quotient of powers
      • Power of a power
      • Zero and negative exponents
    • Square roots and perfect squares
      • Estimating non-perfect square roots
    • Cube roots and higher roots
    • Scientific notation
      • Converting to and from standard form
      • Multiplying and dividing in scientific notation

SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic flashcards — 50 questions with answers

  1. In the number $4{,}728$, what is the place value of the digit $7$?

    It is in the hundreds place, so its value is $7 \times 100 = 700$.

  2. Write the number $6{,}040{,}000$ in expanded form using place values.

    $6{,}000{,}000 + 40{,}000 = (6 \times 10^{6}) + (4 \times 10^{4})$.

  3. What is the rule for adding two integers that have different signs?

    Subtract the smaller absolute value from the larger, and keep the sign of the number with the larger absolute value. Example: $-8 + 3 = -5$.

  4. What is the rule for the sign of a product or quotient of two integers?

    Same signs give a positive result; different signs give a negative result. Example: $(-6)\times(-4)=24$ and $(-6)\div 2 = -3$.

  5. Compute $-7 - (-12)$.

    Subtracting a negative is adding its opposite: $-7 + 12 = 5$.

  6. What does the acronym PEMDAS stand for in order of operations?

    Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

  7. Evaluate $3 + 4 \times 2^{2} - 6 \div 3$ using PEMDAS.

    Exponent: $2^{2}=4$. Then $3 + 4\times 4 - 6\div 3 = 3 + 16 - 2 = 17$.

  8. In order of operations, how do you handle multiplication and division when both appear?

    They have equal priority, so evaluate them from left to right in the order they appear, not multiplication first.

  9. State the commutative, associative, and distributive properties of operations.

    Commutative: $a+b=b+a$ and $ab=ba$. Associative: $(a+b)+c=a+(b+c)$. Distributive: $a(b+c)=ab+ac$.

  10. What is the identity element for addition and the identity element for multiplication?

    The additive identity is $0$ (since $a+0=a$); the multiplicative identity is $1$ (since $a\times 1 = a$).

  11. What is a factor of a number, and what is a multiple of a number?

    A factor divides the number evenly with no remainder (e.g., $4$ is a factor of $12$). A multiple is the product of the number and an integer (e.g., $12$ is a multiple of $4$).

  12. What is a prime number, and what is the only even prime?

    A prime number has exactly two distinct factors: $1$ and itself. The only even prime number is $2$.

  13. Write the prime factorization of $60$.

    $60 = 2^{2} \times 3 \times 5$.

  14. What is the difference between the GCF and the LCM of two numbers?

    The GCF (greatest common factor) is the largest number that divides both. The LCM (least common multiple) is the smallest positive number that both divide into. For $12$ and $18$: GCF $=6$, LCM $=36$.

  15. How do you create a fraction equivalent to a given fraction?

    Multiply (or divide) the numerator and denominator by the same nonzero number. Example: $\frac{2}{3} = \frac{2\times 4}{3\times 4} = \frac{8}{12}$.

  16. How do you simplify a fraction to lowest terms?

    Divide the numerator and denominator by their GCF. Example: $\frac{18}{24} = \frac{18\div 6}{24\div 6} = \frac{3}{4}$.

  17. What is the procedure for adding fractions with unlike denominators?

    Find a common denominator (the LCD), rewrite each fraction with it, add the numerators, and simplify. Example: $\frac{1}{4}+\frac{1}{6}=\frac{3}{12}+\frac{2}{12}=\frac{5}{12}$.

  18. How do you multiply two fractions and how do you divide by a fraction?

    Multiply: numerator times numerator over denominator times denominator. Divide: multiply by the reciprocal of the second fraction. Example: $\frac{2}{3}\div\frac{4}{5}=\frac{2}{3}\times\frac{5}{4}=\frac{10}{12}=\frac{5}{6}$.

  19. How do you convert the mixed number $3\frac{2}{5}$ to an improper fraction?

    Multiply the whole number by the denominator and add the numerator: $\frac{3\times 5 + 2}{5} = \frac{17}{5}$.

  20. When adding or subtracting decimals, what is the key alignment rule?

    Line up the decimal points (and the place values) vertically, filling in zeros as needed, then add or subtract. Example: $2.50 + 1.07 = 3.57$.

Show the other 30 Quantitative Reasoning: Number Sense and Arithmetic flashcards

  1. How do you place the decimal point when multiplying decimals, e.g. $0.3 \times 0.04$?

    Multiply as whole numbers, then give the product the total number of decimal places from both factors. $3\times 4 = 12$ with $1+2 = 3$ decimal places, so $0.3\times 0.04 = 0.012$.

  2. How do you divide by a decimal, e.g. $4.5 \div 0.5$?

    Move the decimal point in the divisor to make it a whole number and shift the dividend's point the same number of places: $45 \div 5 = 9$.

  3. How do you convert a fraction to a decimal?

    Divide the numerator by the denominator. Example: $\frac{3}{8} = 3 \div 8 = 0.375$.

  4. How do you convert a decimal to a percent and a percent to a decimal?

    Decimal to percent: multiply by $100$ (move the point two places right), so $0.45 = 45\%$. Percent to decimal: divide by $100$ (move two places left), so $45\% = 0.45$.

  5. How do you convert a percent to a fraction in lowest terms, e.g. $40\%$?

    Write it over $100$ and simplify: $40\% = \frac{40}{100} = \frac{2}{5}$.

  6. What fraction and decimal are equivalent to $25\%$, $50\%$, and $75\%$?

    $25\% = \frac{1}{4} = 0.25$; $50\% = \frac{1}{2} = 0.5$; $75\% = \frac{3}{4} = 0.75$.

  7. What is the formula to find a percent of a number, and what is $30\%$ of $80$?

    Percent of a number $=$ (percent as a decimal) $\times$ number. $0.30 \times 80 = 24$.

  8. How do you find what percent one number is of another, e.g. what percent is $15$ of $60$?

    Divide the part by the whole and multiply by $100$: $\frac{15}{60}\times 100 = 25\%$.

  9. How do you calculate percent increase or percent decrease?

    $\text{percent change} = \dfrac{\text{amount of change}}{\text{original amount}} \times 100\%$. Example: from $40$ to $50$ is $\frac{10}{40}\times 100\% = 25\%$ increase.

  10. What is a ratio, and what are the three common ways to write the ratio of $3$ to $5$?

    A ratio compares two quantities. It can be written as $3:5$, as $3 \text{ to } 5$, or as the fraction $\frac{3}{5}$.

  11. How do you simplify the ratio $18:24$?

    Divide both terms by their GCF ($6$): $18:24 = 3:4$.

  12. What is a proportion, and what method solves it?

    A proportion states two ratios are equal, $\frac{a}{b}=\frac{c}{d}$. Solve by cross-multiplication: $a\cdot d = b\cdot c$.

  13. Solve the proportion $\frac{x}{12} = \frac{3}{4}$.

    Cross-multiply: $4x = 36$, so $x = 9$.

  14. What is a unit rate, and what is the unit rate for $150$ miles in $3$ hours?

    A unit rate is a rate with a denominator of $1$. $\frac{150 \text{ miles}}{3 \text{ hours}} = 50$ miles per hour.

  15. How do you use unit pricing to find the better buy between two package sizes?

    Divide the price by the quantity for each option to get the price per unit, then choose the lower unit price. Example: $\$4$ for $2$ lb is $\$2/\text{lb}$; $\$9$ for $5$ lb is $\$1.80/\text{lb}$, so the $5$ lb is the better buy.

  16. State the relationship among speed, distance, and time.

    $\text{distance} = \text{speed} \times \text{time}$, which rearranges to $\text{speed} = \dfrac{\text{distance}}{\text{time}}$ and $\text{time} = \dfrac{\text{distance}}{\text{speed}}$.

  17. A car travels $180$ miles at $60$ mph. How long does the trip take?

    $\text{time} = \dfrac{\text{distance}}{\text{speed}} = \dfrac{180}{60} = 3$ hours.

  18. What is a scale (or scale factor) in a scale drawing or map?

    It is the ratio of a length in the drawing to the corresponding actual length, e.g. $1 \text{ inch} : 50 \text{ miles}$, telling how much the real object has been reduced or enlarged.

  19. On a map with scale $1 \text{ inch} = 25$ miles, how far apart are two cities that are $4$ inches apart?

    Set up a proportion: $\frac{1}{25}=\frac{4}{x}$, so $x = 100$ miles.

  20. State the product and quotient rules for exponents with the same base.

    Product: $a^{m}\cdot a^{n} = a^{m+n}$. Quotient: $\dfrac{a^{m}}{a^{n}} = a^{m-n}$.

  21. State the power-of-a-power rule and the value of any nonzero base raised to the zero power.

    Power of a power: $(a^{m})^{n} = a^{mn}$. Zero exponent: $a^{0} = 1$ for any $a \neq 0$.

  22. What does a negative exponent mean, e.g. $2^{-3}$?

    A negative exponent means the reciprocal of the positive power: $2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}$.

  23. What is a square root, and what is $\sqrt{49}$?

    A square root of a number is a value that, when multiplied by itself, gives that number. $\sqrt{49} = 7$ because $7^{2} = 49$.

  24. List the perfect squares from $1$ to $144$.

    $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$ (the squares of $1$ through $12$).

  25. Between which two consecutive whole numbers does $\sqrt{50}$ lie?

    Between $7$ and $8$, because $7^{2}=49$ and $8^{2}=64$, and $50$ is between them (closer to $7$).

  26. How do you simplify $\sqrt{72}$?

    Factor out the largest perfect square: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}$.

  27. What is a cube root, and what is $\sqrt[3]{27}$?

    A cube root of a number is a value that, multiplied by itself three times, gives that number. $\sqrt[3]{27} = 3$ because $3^{3} = 27$.

  28. List the perfect cubes from $1$ to $125$.

    $1, 8, 27, 64, 125$ (the cubes of $1, 2, 3, 4, 5$).

  29. Evaluate $\sqrt[3]{-64}$ and explain why a cube root of a negative number is defined.

    $\sqrt[3]{-64} = -4$ because $(-4)^{3} = -64$. Odd roots of negative numbers are real because an odd number of negative factors stays negative.

  30. What does the index $n$ in the radical $\sqrt[n]{x}$ tell you, and what is $\sqrt[4]{81}$?

    The index $n$ tells which root to take (what power gives $x$). $\sqrt[4]{81} = 3$ because $3^{4} = 81$.

SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic FAQ

What is in the SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic syllabus?

SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic is split into 4 chapters: Integers and the Number System, Fractions, Decimals, and Percents, Ratios, Proportions, and Rates and Powers, Roots, and Scientific Notation. Together they contain 19 topics and 37 sub-topics.

How many chapters are in SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic?

4 chapters, containing 19 topics and 37 sub-topics. The largest chapter is Integers and the Number System, with 5 topics.

Are there flashcards for SSAT (Secondary School Admission Test) Quantitative Reasoning: Number Sense and Arithmetic?

Yes — a 50-card Quantitative Reasoning: Number Sense and Arithmetic deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.