← ISEE® Middle Level

Section 2: Quantitative Reasoning

Thirty-seven items in thirty-five minutes where every wrong answer is a logical error, not a slip: word problems plus quantitative comparisons.

23 topics · 97 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.

What Quantitative Reasoning Actually Measures

  1. How it worksHow does invoking a rule and then applying it differ from grinding out an arithmetic answer?
  2. Common mistakeWhy isn't this just an easier version of the arithmetic already done in class every day?
  3. Worked exampleFind how many green apples a basket holds when 40 percent of it is 6 red apples and the rest are green.
  4. Going furtherWhat changes when a reasoning item hands you a rule you have never met and asks you to apply it once?
Your turneasy1 of 3
A basket holds 70 apples and 40 percent of them are red. The rest are green. How many green apples are in the basket?

Logical Errors vs Process Errors

  1. How it worksHow are reasoning wrong answers built from faulty logic while achievement wrong answers come from faulty procedure?
  2. Worked exampleWalk me through how a student who computes 40 percent of 6 instead of treating 6 as 40 percent lands on a trap choice.
  3. Common mistakeWhy doesn't checking your arithmetic twice protect you on a reasoning item?
  4. Going furtherWhat happens to your error pattern the moment you cross from a reasoning section into an achievement section?
Your turneasy1 of 3
On item 22 of Quantitative Reasoning, how are the three wrong choices built?

Sorting Out Irrelevant Information

  1. How it worksHow do you decide which numbers in a word problem actually feed the quantity being asked for?
  2. Worked exampleFind the cost of 4 notebooks at 1.25 dollars each when the problem also says the shop opened in 1998 and stocks 312 pens.
  3. Common mistakeWhy isn't every number printed inside a word problem part of the calculation?
  4. Going furtherWhat changes when the spare detail is a unit of measure rather than a quantity?
  5. Ties togetherHow does crossing out spare detail connect to reading a chart that plots more series than one question needs?
Your turneasy1 of 3
A shop that opened in 1997 and stocks 312 pens sells notebooks at 7 dollars each. Write an expression for the cost, in dollars, of n notebooks.

Estimation and Mental Math Without a Calculator

  1. How it worksHow does rounding both numbers to friendly benchmarks pick an answer without ever finding it exactly?
  2. Worked exampleFind which is larger, 48 times 5.2 or 240, by rounding rather than multiplying out.
  3. Common mistakeWhy doesn't rounding both numbers upward ever give a safe comparison?
  4. Going furtherWhat happens to an estimate when two numbers are nudged the same direction by very different amounts?
Your turneasy1 of 3
A student estimates 48 times 5.2 by rounding 5.2 down to 5. By how much does her estimate fall short of the true product?

Choosing the Operation a Situation Calls For

  1. How it worksHow do the words per, each, shared and altogether signal which operation a situation needs?
  2. Worked exampleWalk me through deciding whether 144 seats in 8 equal rows calls for 144 divided by 8 or 144 times 8.
  3. Common mistakeWhy isn't the word more always a signal to add?
  4. Worked exampleFind how many 0.75 litre bottles a 12 litre tank fills, and show why that is division.
  5. Going furtherWhat changes about the operation when the amount being shared out is a fraction smaller than one?
Your turneasy1 of 3
A tank holds 23 litres and is filled from bottles of 0.25 litres each. How many bottles does it take?

Pacing 37 Questions in 35 Minutes

  1. How it worksHow does 57 seconds per quantitative item compare with the 51 seconds each achievement item gets?
  2. Worked exampleFind how many minutes remain for the last 10 quantitative items when the first 27 took 28 minutes.
  3. Common mistakeWhy doesn't marching through every item in printed order beat skipping the hard ones and returning?
Your turneasy1 of 3
Quantitative Reasoning has 37 items in 35 minutes. A student has finished 20 of them and has 19 minutes left. How many seconds does each remaining item get?

Why You Never Leave a Blank

  1. How it worksHow does a blind guess among four choices turn into an expected quarter of a mark?
  2. Worked exampleFind the expected gain from guessing on the last 8 items nobody had time to reach.
  3. Common mistakeWhy isn't an empty bubble safer than risking a wrong one on a scored section?
Your turneasy1 of 3
A student runs out of time with 15 items unreached and guesses blindly on all of them. Each has four choices. How many marks does she expect to gain?

The Quantitative Comparison Format

  1. How it worksHow do four fixed choices cover every possible relationship two quantities could have?
  2. Worked exampleCompare 50 percent of 60 against 30 percent of 100 and pick among greater, greater, equal or undetermined.
  3. Common mistakeWhy isn't the fourth choice simply what you pick once the clock runs low?
  4. Going furtherWhat changes when a comparison prints expressions holding a letter instead of two plain numbers?
  5. Ties togetherHow does a comparison with no numbers at all connect to solving an inequality rather than an equation?
Your turneasy1 of 3
Quantity A: 50 percent of 70. Quantity B: 5 percent of 700. Which statement is true?

Comparing Without Computing

  1. How it worksHow does cancelling identical pieces off both sides leave a relationship you can read instantly?
  2. Worked exampleCompare 37 times 41 against 41 times 38 without multiplying either product.
  3. Common mistakeWhy doesn't a comparison ever require the actual value of either quantity?
  4. Worked exampleCompare 5/9 against 5/8 without turning either fraction into a decimal.
  5. Going furtherWhat happens to the relationship when both quantities are multiplied by the same negative number?
Your turneasy1 of 3
Quantity A: 44 times 41. Quantity B: 41 times (44 plus 1). Which statement is true?

Simplifying Both Columns in Parallel

  1. How it worksHow does taking the same amount off both sides leave the relationship between them untouched?
  2. Worked exampleCompare 3n plus 14 against 2n plus 14 for positive whole n by stripping the shared 14 off both.
  3. Common mistakeWhy isn't dividing both sides by a letter as safe as subtracting a number from both?
  4. Going furtherWhat changes about parallel simplifying when one side holds a square and the other a square root?
Your turneasy1 of 3
n is a positive whole number. Quantity A: 3n plus 22. Quantity B: 2n plus 22. Which statement is true?

Testing Strategic Values

  1. How it worksHow do five probes, zero, one, a negative, a fraction under one and a huge number, cover the whole number line?
  2. Worked exampleCompare n squared against n by trying n equal to 2, then n equal to one half, then n equal to negative 3.
  3. Common mistakeWhy doesn't one successful probe prove that a quantity is always the larger one?
  4. Going furtherWhat happens to a comparison of n against n cubed the moment n is allowed to sit between zero and one?
  5. Ties togetherHow does probing with chosen numbers connect to graphing two curves and watching where they cross?
Your turnmedium1 of 3
n is a number. Quantity A: n to the power 3. Quantity B: n. Which statement is true?

When Cannot Be Determined Is Right

  1. How it worksHow does one probe where the first wins and another where the second wins prove no order exists?
  2. Worked exampleCompare x against 0.5 knowing only that 4 plus x rounds to 5 at the nearest whole number.
  3. Common mistakeWhy isn't an unstated letter automatically a signal that nothing can be decided?
Your turnmedium1 of 3
7 plus x rounds to 7 plus 1 at the nearest whole number. Quantity A: x. Quantity B: 0.5. Which statement is true?

When Cannot Be Determined Is a Trap

  1. How it worksHow does a shared condition pin an order down even when neither quantity can be found alone?
  2. Worked exampleCompare x against y knowing 5x equals 4y and both are positive, without finding either number.
  3. Common mistakeWhy doesn't a letter sitting in both quantities make the fourth choice the safe default?
  4. Worked exampleCompare the perimeter of a square of area 36 against 24 when no side length is stated anywhere.
  5. Going furtherWhat happens when an order fails for every value except the ones the problem actually allows?
Your turnmedium1 of 3
x and y are positive numbers with 8x equal to 3y. Quantity A: x. Quantity B: y. Which statement is true?

Comparisons Built on a Shared Condition

  1. How it worksHow do you carry a shared equation into both quantities instead of solving it first?
  2. Worked exampleCompare the value of x when y is 15 against the value of y when x is 4, for the rule y equals 4x minus 1.
  3. Worked exampleCompare x against y when x divided by 4 plus 5 equals 25 and 6y plus 11 equals 77.
  4. Common mistakeWhy isn't the condition attached to a comparison just decoration you may skim past?
  5. Going furtherWhat changes when the attached condition is an inequality rather than an equation?
Your turnmedium1 of 3
The rule is y equal to 4x minus 1. Quantity A: the value of x when y is 25. Quantity B: the value of y when x is 7. Which statement is true?

Geometry and Measurement Comparisons

  1. How it worksHow does scaling a figure by 3 multiply its area by 9 and its volume by 27?
  2. Worked exampleCompare the area of a triangle of base 2 and height 8 against a triangle of base 3 and height 6.
  3. Common mistakeWhy doesn't a figure drawn bigger on the page have to be bigger in the measurement being compared?
Your turnmedium1 of 3
Quantity A: the area of a triangle with base 2 and height 17. Quantity B: the area of a triangle with base 3 and height 6. Which statement is true?

Ratio and Proportion Reasoning

  1. How it worksHow does a part-to-part ratio become a part-to-whole fraction once the parts are added?
  2. Worked exampleFind how many boys sit in a class of 30 when boys to girls run 2 to 3.
  3. Common mistakeWhy isn't a ratio of 2 to 3 the same claim as two thirds of the group?
  4. Going furtherWhat changes about a ratio when the same number is added to both of its parts?
  5. Ties togetherHow does scaling a ratio connect to enlarging a scale drawing by that same factor?
Your turneasy1 of 3
A recipe uses 8 cups of flour for every 3 cups of milk. Write an expression for the cups of flour needed with x cups of milk.

Percent Reasoning with Benchmarks

  1. How it worksHow do the benchmarks 10 percent, 5 percent and 50 percent build almost any percent by stacking pieces?
  2. Common mistakeWhy doesn't 15 percent of 40 come out any different from 40 percent of 15?
  3. Worked exampleFind 35 percent of 240 using nothing but ten-percent and five-percent chunks.
Your turneasy1 of 3
Find 35 percent of 700, using ten-percent and five-percent chunks rather than multiplying.

Reading Graphs and Drawing Conclusions

  1. How it worksHow do you read a trend off a line graph without reading any single value on it?
  2. Worked exampleFind which quantity grew faster when one rose from 20 to 60 over 4 years and another from 5 to 35 over the same span.
  3. Common mistakeWhy doesn't a steeper-looking climb always mean a faster rate when two vertical scales differ?
  4. Going furtherWhat happens to a conclusion when the vertical axis of a bar chart starts at 90 instead of zero?
  5. Ties togetherHow does reading a circle graph connect to turning a percent into a slice of 360 degrees?
Your turneasy1 of 3
A line on a graph climbs from 20 to 68 over 4 years. By how much does it rise each year on average?

Probability of an Event by Reasoning

  1. How it worksHow does counting favourable outcomes over total outcomes turn a chance into a fraction?
  2. Worked exampleFind the chance of drawing red from a bag of 4 red, 5 blue and 3 green marbles.
  3. Common mistakeWhy isn't a 1 in 6 chance a promise of one six within six rolls?
  4. Going furtherWhat happens to the second draw when the first marble never goes back in the bag?
Your turneasy1 of 3
A bag holds 4 red marbles, 10 blue and 3 green. What is the chance of drawing a red one, as a decimal?

Measurement Concepts and Sensible Units

  1. How it worksHow does the size of a unit decide whether the count climbs or falls when converting?
  2. Common mistakeWhy doesn't a bigger count of units have to mean a bigger measurement?
  3. Worked exampleFind which is longer, 250 centimetres or 2 metres, with no conversion factor written down.
  4. Going furtherWhat changes when a rate welds two units together, like metres per second against kilometres per hour?
Your turneasy1 of 3
One rope is 310 centimetres long and another is 2 metres long. What is the difference between them, in centimetres?

Arriving at a Statistical Solution

  1. How it worksHow does an average behave like a balance point while the middle value behaves like a marker?
  2. Worked exampleFind the mean and the median of 4, 6, 7, 8 and 95, then say which one describes the group.
  3. Common mistakeWhy isn't the average always the number that best stands in for a set?
  4. Going furtherWhat happens to an average when every value in the set is raised by 5?
  5. Ties togetherHow does an average sliding toward an outlier connect to a bar chart carrying one giant bar?
Your turneasy1 of 3
Find the mean of 4, 6, 7, 8 and 102.

Patterns and Rule Generation

  1. How it worksHow does the jump between neighbouring terms turn into a rule good for any position?
  2. Worked exampleFind the 20th term of 4, 7, 10, 13 without writing out every term in between.
  3. Common mistakeWhy doesn't a rule that fits the first three terms have to fit the fourth?
  4. Worked exampleFind the rule behind 2, 6, 18, 54 and use it to reach the sixth term.
  5. Going furtherWhat changes about the rule when the gaps between terms keep growing instead of staying fixed?
Your turneasy1 of 3
A sequence runs 7, then 7 plus 3, then 7 plus 6, then 7 plus 9, and carries on the same way. Write an expression for term number n.

Working Backwards from the Answer Choices

  1. How it worksHow does the numerical ordering of ISEE choices let a single test throw away half of them?
  2. Worked exampleFind which of 12, 15, 18 and 24 leaves remainder 3 after division by 7, by testing each one.
  3. Common mistakeWhy doesn't backsolving help on an item asking which statement must always be true?
Your turneasy1 of 3
A whole number leaves a remainder of 3 when it is divided by 7, and the quotient is 11. What is the number?

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