Floor, Ceiling and Fractional Part in CAT: 4 Patterns

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The floor, ceiling and fractional part questions in CAT look odd the first time you meet them. Brackets, braces and strange symbols sit around an ordinary expression, and it is not obvious what to do. Once you know what each symbol means, most of these questions turn out to be short.

There are four patterns that cover almost everything: converting a floor into an interval, handling negative numbers, working with the fractional part, and counting with floors. This guide shows each with numbers you can check by hand.

What the Symbols Mean

The floor of x, written with brackets that close at the bottom, is the greatest integer that is less than or equal to x. The floor of 3.7 is 3, and the floor of 5 is 5. The ceiling of x, written with brackets that close at the top, is the smallest integer that is greater than or equal to x. The ceiling of 3.7 is 4, and the ceiling of 5 is 5.

The fractional part of x, usually written with curly braces, is x minus the floor of x. For 3.7 it is 0.7. For 5 it is 0. By definition it is never negative and never reaches 1, so it always lies from 0 up to but not including 1. That bound is the single most useful fact in this chapter.

You can always write a number as its floor plus its fractional part. That decomposition, an integer part and a part below 1, lets you split many equations into an integer problem and a small range problem. Most of the patterns below use that split in one form or another.

Pattern One: Turn a Floor Into an Interval

The most common question gives an equation with a floor and asks for the range or number of solutions. The key rule is this: the floor of an expression equals n exactly when the expression lies from n up to but not including n plus 1.

Take the floor of (2x plus 1) equal to 5. By the rule, 5 is at most 2x plus 1, and 2x plus 1 is less than 6. Subtract 1 to get 4 at most 2x less than 5, and divide by 2 to get 2 at most x less than 2.5. So the solution is an interval, from 2 up to but not including 2.5.

Notice that the answer is a range, not a single value. Students expecting one number sometimes force an integer. If the question asks for integer solutions, count the integers in the interval. Here, only x equal to 2 qualifies.

For ceilings, the rule shifts by one: the ceiling of an expression equals n exactly when the expression is greater than n minus 1 and at most n. The open end moves to the left. Write the pair of inequalities down every time, because mixing up which end is open is the most common slip.

Statement	Interval for the expression
Floor of E equals n	n at most E, and E less than n plus 1
Ceiling of E equals n	n minus 1 less than E, and E at most n
Fractional part of E	From 0 up to but not including 1
Floor of E at least n	E at least n

Pattern Two: Negative Numbers

Negative numbers are where the symbols bite. The floor of minus 3.7 is minus 4, because minus 4 is the greatest integer not above minus 3.7. The ceiling of minus 3.7 is minus 3. The fractional part of minus 3.7 is minus 3.7 minus minus 4, which is 0.3, and not 0.7.

A handy identity connects the two functions. The ceiling of x equals minus the floor of minus x. Test it with x equal to 3.7: the floor of minus 3.7 is minus 4, and minus that is 4, which matches the ceiling of 3.7. This identity lets you convert any ceiling question into a floor question when that is easier.

Another useful fact: the floor of x plus the floor of minus x equals 0 when x is an integer, and minus 1 otherwise. Check with x equal to 2.5: the floor is 2, and the floor of minus 2.5 is minus 3, so the sum is minus 1. Questions that sum a floor with a floor of the negative are built on this.

Pattern Three: The Fractional Part as a Bounded Variable

The fractional part is a variable you know a lot about: it lies from 0 up to but not including 1. When an equation contains the fractional part, use that bound to limit the possible integer parts.

Suppose a number x has floor equal to 3 and fractional part equal to one tenth of its floor. Then the fractional part is 0.3, and x equals 3.3. This kind of question can be solved in one line once the decomposition is written down.

A harder version gives an equation such as x equals 4 times the fractional part plus an integer. Write x as n plus f, where n is an integer and f is the fractional part. Substitute, solve for f in terms of n, and then use the bound that f lies from 0 up to but not including 1 to find which integers n are possible. The bound turns an infinite family into a short list.

This is a general method. Introduce the integer part and the fractional part as separate unknowns, solve the algebra, then apply the range on the fractional part to filter the answers. It is the same idea as splitting a remainder problem into a quotient and a remainder.

Pattern Four: Counting With Floors

Floors also count things. The number of multiples of 7 up to 100 is the floor of 100 divided by 7, which is 14, since 98 is the last multiple. In general, the number of multiples of k up to N is the floor of N divided by k.

The same idea gives the highest power of a prime in a factorial. For the power of 2 in 20 factorial, add the floors of 20 over 2, 20 over 4, 20 over 8 and 20 over 16. That is 10 plus 5 plus 2 plus 1, or 18. Each term counts the numbers divisible by 2, by 4, by 8 and by 16, and each higher power adds one more factor for those numbers.

Sums of floors also appear. The sum of the floor of k over 3 for k from 1 to 10 is 0 plus 0 plus 1 plus 1 plus 1 plus 2 plus 2 plus 2 plus 3 plus 3, which is 15. Grouping in blocks of three, where the floor stays constant, is faster than computing each term. Look for the repeat length, which is the denominator.

These counting uses link directly to the number system chapter, where factorials and divisibility questions are common. After you have the pattern, practise number system questions that ask for the power of a prime in a factorial or the number of multiples in a range.

Which Pattern Is the Question Asking For?

A quick way to sort a floor question is to look at what it asks for. If it gives an equation and asks for x, you are in pattern one. If the numbers are negative, pattern two applies. If a curly brace appears, you are splitting into an integer and a fractional part. If the question asks how many, how many multiples, or what power, you are counting.

Many questions combine two patterns. A question may convert a floor into an interval and then count the integers in it. The way to stay organised is to write each step separately: first the interval, then the integers inside it, then the count. Short steps are less error prone than a single leap.

It also helps to test the answer. Substitute one value from your final interval into the original equation and check that it works. Floor questions are friendly to testing, because the numbers are usually small, and a thirty second check catches the open and closed end mistakes.

How Much Time Does This Deserve?

Floor and ceiling questions are a small corner of Quant. Historically, Arithmetic and Algebra have carried more of the section, so treat this as a topic to sequence by return on effort. A short block of practice is enough to make the four patterns automatic, and then you can park the rare harder variants.

A sensible plan is two or three short sessions. The first covers the definitions and pattern one. The second covers negatives and the fractional part. The third covers counting and mixed questions. Our list of the Quant topics that carry the most weight can help you place this chapter against the others.

If you prefer a formula sheet to look at before practice, the free CAT cheatsheets page is a handy place to check your notes against. And a CAT study planner can slot these short sessions into the weeks you have left.