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Quantitative Reasoning: Arithmetic

The whole ETS arithmetic list, from primes and remainders through ratios, percent change and sequences - the content half of the quant sections.

30 topics · 119 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Factors, Multiples and Divisibility

  1. How it worksHow do you find every positive factor of 60 without missing one, by pairing them off?
  2. Worked exampleFind all twelve positive factors of 60 by walking up from 1 and pairing each with its partner.
  3. Common mistakeWhy isn't 0 a factor of any integer, even though 0 is a multiple of every integer?
  4. Worked exampleFind whether 3,417 is divisible by 3 and by 9 using nothing but the sum of its digits.
Your turneasy1 of 3
The integer N equals 25 times 34. How many positive factors does N have?

Prime Numbers and Prime Factorization

  1. How it worksHow does a factor tree break 1,155 down until only primes are left on the branches?
  2. How it worksWalk me through why an integer's prime factorization is unique no matter which split you make first.
  3. Worked exampleFind the prime factorization of 800 and write it out using exponents.
  4. Common mistakeWhy isn't 1 counted as a prime number when nothing divides it except itself?
  5. Going furtherWhat happens to the count of a number's factors when you read the exponents in its prime factorization?
Your turneasy1 of 3
Find the number of positive factors of 26 times 54 times 7.

Greatest Common Divisor and Least Common Multiple

  1. How it worksHow do the prime factorizations of 30 and 75 hand you their gcd and their lcm at the same time?
  2. Common mistakeWhy isn't the product of two numbers the same as their least common multiple?
  3. Worked exampleFind the greatest common divisor and the least common multiple of 30 and 75.
  4. Going furtherWhat happens to the gcd and the lcm when two numbers share no prime factor at all?
Your turnmedium1 of 3
Find the greatest common divisor of 12 times 11 and 18 times 11.

Quotients and Remainders

  1. How it worksHow does the rule that a remainder sits between 0 and the divisor pin the quotient down?
  2. Worked exampleFind the quotient and the remainder when 100 is divided by 45.
  3. Worked exampleFind the quotient and the remainder when negative 32 is divided by 3.
  4. Common mistakeWhy isn't the remainder equal to negative 3 when negative 13 is divided by 5?
  5. Going furtherWhat happens to the remainder pattern as you keep adding the divisor to the number being divided?
Your turneasy1 of 3
Let N = 12 times 14 plus 5, where 5 is smaller than 12. Find the quotient when N is divided by 12.

Odd and Even Integers

  1. How it worksHow do the six odd-even rules fall out of blocks of two that either pair up or leave one over?
  2. Worked exampleFind whether 3 times an odd integer plus 4 times an even integer comes out odd or even.
  3. Common mistakeWhy isn't zero an odd number when it sits squeezed between two odd numbers on the line?
Your turnmedium1 of 3
The integer n equals 2 times 21 plus 1. Find the value of (n2 + n) divided by 2.

Fractions: Equivalence and Comparison

  1. How it worksHow does multiplying top and bottom by the same number leave a fraction's value untouched?
  2. Common mistakeWhy isn't the fraction with the bigger denominator always the smaller of the two fractions?
  3. Worked exampleCompare 7 over 9 with 5 over 7 by cross multiplying rather than finding a common denominator.
Your turneasy1 of 3
Quantity A is the fraction 8 over 11. Quantity B is the fraction 3 over 10. All four values are positive integers as drawn.

Adding, Subtracting, Multiplying and Dividing Fractions

  1. How it worksHow does a common denominator turn two different-sized pieces into pieces you can actually count?
  2. How it worksWalk me through why dividing by a fraction is the same as multiplying by its flip.
  3. Worked exampleFind 3 over 4 minus 5 over 6, then divide that result by 2 over 9.
  4. Common mistakeWhy isn't 1 over 2 plus 1 over 3 equal to 2 over 5?
Your turneasy1 of 3
Find the value of 9 over 7 minus 4 over 11.

Exponents: Bases, Powers and Signs

  1. How it worksHow does writing 3 to the 4th as four copies of 3 multiplied together make every exponent rule obvious?
  2. Worked exampleFind the value of negative 3 squared and of the quantity negative 3, all squared, and show why they differ.
  3. Common mistakeWhy isn't negative 2 raised to the 4th power equal to negative 16?
  4. Going furtherWhat happens to a number between 0 and 1 as you raise it to higher and higher powers?
Your turnmedium1 of 3
Find the value of (-5)4 minus -54.

Zero and Negative Exponents

  1. How it worksHow does walking down the powers of 2 and halving each time force 2 to the zero to be 1?
  2. Worked exampleFind the value of 5 raised to the negative 3 power and write it as a fraction.
  3. Going furtherWhat happens when you raise a fraction such as 2 over 3 to a negative exponent?
Your turneasy1 of 3
Find the value of 70 + 7−2.

Square Roots and the Rules for Radicals

  1. How it worksHow do the rules for splitting a root across a product and across a quotient actually work?
  2. Going furtherWhat happens when a GRE question asks for the square root of a negative number?
  3. Worked exampleFind the simplified form of the square root of 24 plus the square root of 54.
  4. Common mistakeWhy isn't the square root of 9 plus 16 the same as the square root of 9 plus the square root of 16?
Your turnmedium1 of 3
Find the value of √(72 times 8) times √8.

Higher-Order Roots and Fractional Exponents

  1. How it worksHow does writing a root as a fractional exponent turn every radical rule into an exponent rule?
  2. Worked exampleFind the value of 27 raised to the two thirds power without touching a calculator.
  3. Common mistakeWhy isn't 2 the only real number whose fourth power comes out as 16?
  4. Going furtherWhat happens to the cube root of a number when that number is multiplied by 8?
Your turneasy1 of 3
Find the value of (93) raised to the two thirds power.

Arithmetic Operations and Order of Operations

  1. How it worksHow does order of operations resolve an expression holding parentheses, a root, a product and a sum?
  2. Common mistakeWhy isn't multiplication always done before division in an expression that contains both?
  3. Worked exampleFind the value of 6 plus 12 divided by 3 times 2 minus 4 squared.
Your turnmedium1 of 3
Find the value of 7 + 35 / 6·2 - 32.

Decimal Place Value and Powers of Ten

  1. How it worksHow does writing 7,532.418 as a sum of powers of ten expose what each digit is worth?
  2. Worked exampleFind the fraction equal to 0.612 by reading off its place values.
  3. Going furtherWhat happens to a decimal when you multiply it by 10 raised to the 3rd power?
Your turneasy1 of 3
Find the value of 0.65 times 104.

Terminating Decimals, Repeating Decimals and Irrational Numbers

  1. How it worksHow does long division force a fraction's digits to either stop or start repeating?
  2. Worked exampleFind the decimal form of 25 over 12 and mark which of its digits repeat.
  3. Common mistakeWhy isn't every decimal that never ends an irrational number?
  4. Common mistakeWhy doesn't the square root of 2 ever settle into a repeating block of digits?
  5. Going furtherWhat happens to a fraction's decimal when its denominator has a prime factor other than 2 or 5?
Your turnmedium1 of 3
Find the sum of the first 9 digits after the decimal point in the decimal expansion of 1 divided by 3.

The Real Number Line and Ordering

  1. How it worksHow does a number's position on the line decide which of two numbers is the smaller one?
  2. Worked exampleCompare negative 3 halves, negative 1.4 and negative the square root of 2 by placing them on a line.
  3. Common mistakeWhy isn't negative 5 greater than negative 2 just because 5 is greater than 2?
  4. Worked exampleFind which of negative 4, 0, 5 and 10 sits farthest from 1 on the number line.
Your turneasy1 of 3
Quantity A is -4. Quantity B is -14.

Intervals on the Number Line

  1. How it worksHow does a double inequality like 2 less than x less than 3 pick out a stretch of the line?
  2. Going furtherWhat happens to an interval when only one of its two endpoints is included?
  3. Common mistakeWhy isn't x greater than 4 the same set of numbers as x greater than or equal to 4?
Your turneasy1 of 3
How many integers x satisfy 4 less than x, and x less than or equal to 20?

Absolute Value as Distance

  1. How it worksHow does reading absolute value as distance from zero make every sign question disappear?
  2. How it worksWalk me through why the absolute value of a sum is never bigger than the sum of the absolute values.
  3. Worked exampleFind the distance between negative 7 and 4 on the number line using absolute value.
  4. Common mistakeWhy isn't the absolute value of x always just x with the minus sign stripped off the front?
  5. Going furtherWhat happens to the V-shaped graph of absolute value of x when you change it to absolute value of x minus 3?
Your turneasy1 of 3
Find the value of abs(7 - 17) + abs(17 - 7).

Properties of Real Numbers

  1. How it worksHow does the distributive property turn 5 times 19 into something you can do in your head?
  2. Common mistakeWhy isn't division by zero simply equal to zero, or to some kind of infinity?
  3. Worked exampleFind the value of 5 times the quantity 3 plus 16 in two different ways and show they agree.
  4. Going furtherWhat happens to a product when exactly one of the two numbers being multiplied is negative?
Your turneasy1 of 3
Find the value of 7 times 30 plus 7 times 28, using the distributive property to do a single multiplication.

Estimation and Ballparking

  1. How it worksHow do you round one factor up and the other down so the two errors cancel instead of stacking?
  2. Worked exampleFind an estimate for 350 divided by 33, times 2.95, good enough to choose among 10, 20, 30, 40 and 50.
  3. Going furtherWhat happens to an estimate's reliability when all five answer choices sit within 5 percent of each other?
  4. Ties togetherHow does estimating first catch a key-entry error before it quietly costs you the question?
Your turneasy1 of 3
Find 65 percent of 700.

Ratio: Part-to-Part and Part-to-Whole

  1. How it worksHow does a divided bar show a part-to-part ratio and a part-to-whole fraction at the same time?
  2. Worked exampleFind what fraction of the basket is apples when the apples to oranges ratio is 8 to 12.
  3. Worked exampleFind the reduced ratio when a basket holds 5 apples, 30 pears and 20 oranges.
  4. Common mistakeWhy isn't a ratio of 2 to 3 the same thing as two thirds of the whole?
  5. Going furtherWhat happens to a ratio of 1 to 6 to 4 when every one of the quantities is tripled?
Your turneasy1 of 3
A basket holds apples and oranges in the ratio 7 to 8, and it holds 19 times (7 + 8) pieces of fruit in all. Find the number of apples.

Proportions and Combining Ratios

  1. How it worksHow does cross multiplication turn a proportion into a plain linear equation?
  2. How it worksWalk me through combining a ratio of A to B with a ratio of B to C when the shared term does not match.
  3. Worked exampleFind x when the ratio of x to 49 is the same as the ratio of 3 to 21.
  4. Common mistakeWhy isn't the ratio of A to C found by simply multiplying the two given ratios together?
Your turneasy1 of 3
Find x when the ratio of x to 48 equals the ratio of 7 to 11.

Percent, Fraction and Decimal Equivalents

  1. How it worksHow does a hundred-square grid turn 32 percent into the fraction 32 over 100 and the decimal 0.32?
  2. Worked exampleFind the decimal and fraction forms of 0.3 percent, of 8 percent and of 250 percent.
  3. Common mistakeWhy isn't 0.01 the same number as 0.01 percent?
Your turneasy1 of 3
Find the decimal equal to 175 percent.

Finding the Part and Finding the Whole

  1. How it worksHow does the part over whole equals percent over 100 proportion solve all three of those shapes?
  2. Worked exampleFind what percent of 150 the number 12.9 is.
  3. Common mistakeWhy isn't 20 percent of 50 a different number from 50 percent of 20?
  4. Going furtherWhat happens when a problem hands you the part and the rate and asks you for the whole?
Your turnmedium1 of 3
Find what percent of 154 the number 45 is.

Percent Increase and Percent Decrease

  1. How it worksHow does the base of a percent change decide the answer, and why is it always the starting amount?
  2. Common mistakeWhy isn't a rise from 150 to 300 a 200 percent increase?
  3. Worked exampleFind the value of a 1,300 dollar investment after it gains 12 percent in a single day.
  4. Going furtherWhat happens to a percent decrease as the quantity falls all the way down to zero?
  5. Ties togetherHow does a percent increase become a single multiplier you can reuse on any starting amount?
Your turneasy1 of 3
Find the value of a 1545 dollar investment after it gains 18 percent in a single day.

Successive Percent Changes

  1. How it worksHow does each percent change reset the base so the second change measures a different amount?
  2. How it worksWalk me through folding two successive changes into one multiplication by 0.92 and then by 1.06.
  3. Worked exampleFind the overall percent change when enrollment drops 8 percent one year and rises 6 percent the next.
  4. Common mistakeWhy isn't a 10 percent rise followed by another 10 percent rise the same as a 20 percent rise?
  5. Going furtherWhat happens to the combined effect when the same percent change is applied three times running?
Your turnhard1 of 3
Enrollment drops 15 percent one year and rises 13 percent the next. Find the overall percent change, counting an increase as positive.

Percent of a Percent and Percentage Points

  1. How it worksHow does taking a percent of a percent shrink one bar twice instead of only once?
  2. Going furtherWhat happens when a rate moves from 4 percent to 6 percent, measured in points versus measured as change?
  3. Worked exampleFind 30 percent of 20 percent of 5,000.
  4. Common mistakeWhy isn't a jump from 4 percent to 6 percent a 2 percent increase?
Your turnmedium1 of 3
Find 59 percent of 62 percent of 8526.

Rates: Distance, Speed and Time

  1. How it worksHow does distance equals rate times time let you solve for whichever of the three is missing?
  2. Worked exampleFind the average speed for a 120 mile drive that takes 2 hours out and 3 hours back.
  3. Common mistakeWhy isn't the average of 40 and 60 miles per hour the average speed for a there-and-back trip?
  4. Worked exampleFind how many feet per second 6 miles per hour works out to, using the two conversion factors.
Your turnmedium1 of 3
A driver covers 144 miles in 3 hours going out and takes 8 hours coming back along the same route. Find the average speed in miles per hour for the whole trip.

Work and Combined Rates

  1. How it worksHow does adding rates rather than adding times give the right answer for two machines together?
  2. Worked exampleFind how long machines A and B take together when A alone needs 10 minutes and B alone needs 15.
  3. Going furtherWhat happens to the combined time when one worker empties the tank while the other fills it?
  4. Ties togetherHow does the combined-rate setup turn out to be the same move as adding two unlike fractions?
Your turnmedium1 of 3
Machine A alone finishes a job in 16 minutes and machine B alone finishes it in 24 minutes. Find the number of minutes the two take working together.

Arithmetic Sequences

  1. How it worksHow does the common difference hand you the 40th term without listing the first 39?
  2. Worked exampleFind the 40th term of the sequence starting 7, 11, 15, 19 and adding 4 each time.
  3. Going furtherWhat happens to the sum of an arithmetic sequence when you pair the first term with the last?
Your turneasy1 of 3
An arithmetic sequence starts at 11 and adds 7 at each step. Find term number 57.

Geometric Sequences

  1. How it worksHow does the common ratio give you any term of 3, 6, 12, 24 without writing them all out?
  2. Worked exampleFind the 8th term of the geometric sequence that begins 3, 6, 12, 24.
  3. Common mistakeWhy isn't a sequence that grows by larger and larger amounts automatically geometric?
  4. Going furtherWhat happens to a geometric sequence when its common ratio sits between 0 and 1?
Your turnmedium1 of 3
A geometric sequence begins at 7 and multiplies by 3 at each step. Find term number 10.

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