← ISEE® Upper Level

Quantitative Reasoning (37 questions, 35 minutes)

Word problems plus quantitative comparisons, where every wrong answer is a logical error and calculation is often beside the point.

22 topics · 82 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.

Quantitative Reasoning vs Mathematics Achievement

  1. How it worksHow does a Quantitative Reasoning wrong answer come from bad logic while a Mathematics Achievement wrong answer comes from bad arithmetic?
  2. How it worksWalk me through the same idea tested twice, once as a reasoning item and once as an achievement item.
  3. Worked exampleShow me why comparing 40 percent of 80 with 80 percent of 40 is reasoning while computing 40 percent of 80 is achievement.
  4. Common mistakeWhy isn't Quantitative Reasoning just an easier warm-up for the Mathematics Achievement section?
  5. Ties togetherHow do the two math sections get paired against each other on the ISEE score report?
Your turneasy1 of 3
One item asks for 48 percent of 80. Another asks which is greater, 48 percent of 80 or 80 percent of 48. Which one belongs to Quantitative Reasoning?

Rule Generation: Items That Need No Calculation

  1. How it worksHow do you find the rule an item is testing before you touch a single number?
  2. Worked exampleWalk me through why n times m must be a multiple of a times b whenever a divides n and b divides m.
  3. Worked exampleFind the value of (x + y)2 given that x2 + y2 = 40 and 2xy = 24, without ever solving for x or y.
  4. Going furtherWhat changes about your approach when the item hands you the rule inside the question?
Your turneasy1 of 3
It is given that x2 + y2 = 48 and 2xy = 29. What is the value of (x + y)2?

Sorting Relevant From Irrelevant Information

  1. How it worksHow do you decide, before computing anything, which numbers in a word problem actually matter?
  2. Worked exampleFind the muffin revenue when a bakery sells 120 muffins at 2 dollars, 45 scones at 3 dollars, and opens at 6 a.m.
  3. Common mistakeWhy isn't every number printed in an ISEE word problem needed to answer it?
  4. Going furtherWhat happens to a problem's difficulty the moment you cross out the numbers you do not need?
Your turneasy1 of 3
A bakery sells 136 muffins at 4 dollars each and 57 scones at 3 dollars each, and it opens at 6 a.m. What is the muffin revenue, in dollars?

The Quantitative Comparison Format

  1. How it worksHow do the four fixed answer choices work when Column A is set against Column B?
  2. How it worksWalk me through reading the centered information that sits above the two columns before you compare them.
  3. Worked exampleCompare the measure of one interior angle of a regular quadrilateral with one interior angle of a regular hexagon.
  4. Common mistakeWhy isn't the fourth choice, cannot be determined, just there for the questions you find hard?
Your turneasy1 of 3
In a comparison item, Column A holds 31 and Column B holds 66. What are the four answer choices on offer?

Comparing Without Computing

  1. How it worksHow do you strip the same factor off both columns until the comparison becomes obvious?
  2. Worked exampleCompare 37 times 48 with 38 times 47 without multiplying either product out.
  3. Worked exampleCompare 3/7 with 5/11 by cross multiplying instead of converting both to decimals.
  4. Going furtherWhat happens to a cancel-from-both-sides move when the shared factor turns out to be negative?
Your turneasy1 of 3
Column A is 27 times 53. Column B is (27 + 1) times (53 - 1). Which is greater, without multiplying either out?

Testing Strategic Values: 0, 1, a Negative, a Fraction, a Big Number

  1. How it worksHow does dropping 0, 1, a negative, a fraction and a large number into both columns settle a comparison?
  2. Worked exampleCompare x squared with x cubed by testing x = 2, then x = 1/2, then x = 0, then x = -2.
  3. Worked exampleCompare 1/x with x by testing x = 1/2 and then x = 3 and watching the winner flip.
  4. Common mistakeWhy doesn't one value where Column A wins prove that Column A is always the greater one?
Your turneasy1 of 3
Column A is x2 and Column B is x3, with nothing said about x. A student tests x = 7 and finds Column B larger. What has that shown?

When Cannot Be Determined Is the Answer

  1. How it worksHow do two test values that tip the balance opposite ways prove a relationship is undetermined?
  2. Worked exampleCompare 2x with x + 2 when nothing at all is said about x, and show why neither column always wins.
  3. Going furtherWhat happens to that same comparison the instant the item adds that x is greater than 2?
Your turneasy1 of 3
Column A is 2x and Column B is x + 7, with nothing said about x. Which choice is correct?

When Cannot Be Determined Is a Trap

  1. How it worksHow do you tell a genuinely unknown quantity from one that only looks unknown?
  2. Worked exampleCompare the square root of 50 with 7, and show why the answer cannot possibly be undetermined.
  3. Common mistakeWhy isn't I cannot work this out the same thing as the relationship cannot be determined?
  4. Going furtherWhat happens when a comparison holds a variable but every allowed value gives the same verdict?
Your turneasy1 of 3
Column A is the square root of 71 and Column B is 11. Why can the answer not be cannot be determined?

Restricted Domains Stated in the Item

  1. How it worksHow does a stated restriction shrink the set of test values you are allowed to try?
  2. Worked exampleCompare n squared with n when n is a positive integer, then again when n is any real number.
  3. Worked exampleCompare 1/(x - 3) with 0 given only that x is greater than 3.
  4. Common mistakeWhy doesn't the phrase x is a number let you assume x is a whole number?
Your turneasy1 of 3
Write 7n2 - 7n in fully factored form, in terms of n.

The Commonest Comparison Error: Computing Everything

  1. How it worksHow do you spot in two seconds that a comparison is structural rather than arithmetic?
  2. Worked exampleCompare 25 percent of 880 with 88 percent of 250 without finishing either calculation.
  3. Common mistakeWhy isn't grinding out both columns the safe choice when you still have time on the clock?
Your turneasy1 of 3
Column A is 34 percent of 654. Column B is 654 percent of 34. Which is greater?

Variables on Both Sides of the Comparison

  1. How it worksHow do you subtract a matching term from both columns the way you would from both sides of an equation?
  2. Worked exampleCompare 3a + 7 with 2a + 7 + a and show that the two columns are equal for every value of a.
  3. Worked exampleCompare 5y - 4 with 4y - 5 when y is known to be negative.
  4. Going furtherWhat happens to a both-columns move when you are tempted to divide by the variable itself?
Your turneasy1 of 3
Column A is 10a + 8 and Column B is 3a + 8. Write Column A minus Column B in terms of a.

Estimation Instead of Exact Arithmetic

  1. How it worksHow do you round two numbers so your estimate stays on the side of the truth you need?
  2. Worked exampleFind roughly what 19.6 percent of 4,050 is by rounding it to 20 percent of 4,000.
  3. Worked exampleFind which is larger, 612 divided by 29 or the number 21, using estimation only.
  4. Common mistakeWhy doesn't rounding both numbers up always push an estimate in the same direction?
Your turneasy1 of 3
About what is 21 percent of 4095?

Recognizing Which Operation a Situation Calls For

  1. How it worksHow does a word problem's wording tell you whether to divide or to multiply?
  2. Worked exampleFind how many 0.75 liter bottles a 12 liter tank fills, and show why that is division and not multiplication.
  3. Common mistakeWhy isn't the word of always a signal to multiply and the word less always a signal to subtract?
  4. Going furtherWhat changes about the setup when the same situation asks for a rate instead of a total?
Your turneasy1 of 3
How many bottles of 0.76 litres can be filled from a tank holding 24 litres?

Comparing and Contrasting Quantities

  1. How it worksHow do you line two quantities up so that comparing them is actually fair?
  2. Worked exampleCompare a 30 percent discount followed by a 20 percent discount against one single 50 percent discount.
  3. Common mistakeWhy isn't a 20 percent rise followed by a 20 percent fall a return to the original price?
Your turneasy1 of 3
A coat costs 164 dollars. A discount of 33 percent is applied, and then a further discount of 29 percent. What is the final price, in dollars?

Analyzing and Interpreting Data

  1. How it worksHow do you read a data table for the single comparison the question is actually asking for?
  2. Worked exampleFind which class improved most when scores moved 62 to 71, 80 to 86, 45 to 58, and 90 to 94.
  3. Going furtherWhat happens to that conclusion when the question asks for percent change instead of raw change?
Your turneasy1 of 3
A table shows a class mean of 54 in the first year and 74 in the second. What was the percent increase?

Drawing Conclusions From Graphs

  1. How it worksHow do you read a trend off a line graph without reading any individual point?
  2. Worked exampleFind the steepest month when a line graph runs 20, 24, 25, 38, 40 across five months.
  3. Common mistakeWhy doesn't a line that keeps climbing mean the rate of increase is climbing too?
  4. Going furtherWhat changes about a comparison when two lines are drawn on two different scales?
Your turneasy1 of 3
A line graph runs 20, 24, 25, 46 and 40 across five months. Between which months is the line steepest?

Reasoning Out the Probability of an Event

  1. How it worksHow does counting the ways something can fail hand you the probability that it happens?
  2. Worked exampleCompare the chance that two marbles drawn are both yellow with the chance at least one is, from 3 blue, 4 red, 6 green and 10 yellow.
  3. Common mistakeWhy doesn't drawing two marbles without replacing the first give the same odds as replacing it?
Your turneasy1 of 3
A bag holds 3 blue, 4 red, 6 green and 12 yellow marbles. Two are drawn without replacement. What is the probability that both are yellow?

Concepts and Applications of Measurement

  1. How it worksHow does one scale factor act differently on a length, on an area and on a volume?
  2. Worked exampleShow which change doubles a cone's volume, doubling its height or doubling its radius.
  3. Worked exampleFind how the surface area of a cube changes when every edge is multiplied by 3.
  4. Common mistakeWhy isn't a shape with the larger perimeter guaranteed to have the larger area?
Your turneasy1 of 3
Every edge of a rectangular box is multiplied by 4. By what factor does its volume change?

Arriving at a Statistical Solution

  1. How it worksHow do the mean, the median and the range each respond when a whole data set shifts up?
  2. Worked exampleFind the new range after 4 points are added to every score in a set whose range was 35.
  3. Worked exampleFind the mean of 12, 15, 15, 18 and 40, then the median, and show where the gap comes from.
  4. Going furtherWhat happens to the mean when one value is dragged far to the right while the median sits still?
Your turneasy1 of 3
A set of scores has a range of 41. Every score then has 11 points added to it. What is the new range?

Reasoning With Factors and Multiples

  1. How it worksHow does the statement a divides n turn into something you can multiply through?
  2. Worked exampleFind what must be true of nm when 6 divides n and 10 divides m, a multiple of 60 or a factor of 60.
  3. Common mistakeWhy isn't every common multiple of 4 and 6 also a multiple of 24?
Your turneasy1 of 3
It is given that 7 divides n and 13 divides m. Is nm necessarily a multiple of 7 times 13, or a factor of it?

Pacing 37 Items in 35 Minutes

  1. How it worksHow do you budget 35 minutes across 37 items when some of them need no calculation at all?
  2. Worked exampleWalk me through a 35 minute plan spending 40 seconds on each comparison and 70 on each word problem.
  3. Going furtherWhat changes about pacing when 5 of the 37 items are unscored and you cannot tell which ones?
Your turneasy1 of 3
The Quantitative Reasoning section runs 35 minutes. A student spends 44 seconds on each of the 16 quantitative comparisons. How many seconds are left for the rest?

No Penalty, So No Blanks

  1. How it worksHow much does knocking out one of four choices raise what a guess is worth?
  2. Worked exampleFind the expected number correct from guessing on 8 items after eliminating one choice on each.
  3. Common mistakeWhy isn't guessing on the ISEE risky the way it is on a test with a quarter point penalty?
  4. Going furtherWhat changes about your strategy with 90 seconds left and six items still untouched?
Your turneasy1 of 3
A student guesses on 14 items, having eliminated one of the four choices on each of them. How many correct answers should that be expected to produce?

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