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Number, Ranking and Time Sequence Test

Reasoning · Verbal Reasoning

6 Part4 worked examples31 practice questions

This chapter teaches three kinds of questions together, because one thing works in all of them — counting exactly. First: the number test — a long line of digits is given and you must say at how many places a condition is met. Second: the ranking test — at which place a person stands in a queue or class, how many people there are in all, how many are between two persons. Third: time sequence — clock times, days of the week and dates.

There are very few formulas in these questions, but many mistakes, and almost all of them are off-by-one mistakes: counting someone twice, skipping someone, or adding the end persons in “between”. In this chapter we shall keep one check with us everywhere: “is this person included in the count or not?”

Each part first explains the idea, then the method step by step, then a solved example and finally practice. All questions are our own and every answer has been checked by computer.

How to use this chapter

  1. Read — each part first explains the idea, then the method.
  2. Study the solved example — the check of every count is shown at every step.
  3. Try it yourself — draw lines and make marks on paper and then pick an option. The solution shows all the places that were found.
  4. Know the level — Basic, Standard, Advanced.
NoteThe questions you have solved are remembered in this browser. Look at the panel on the right (the “Contents” button on a phone) to see how many you have got right.
1

Number tests

Games with lines of digits and with numbers

Counting by conditions in a digit string, interchanging digits, groups of three-digit numbers and forming numbers from digits.

Counting by conditions in a digit string

“Immediately before” and “immediately after” — slide a window of three digits

The window method for questions with before-and-after conditions, and the cares about overlaps, ends and “and / or”.

In this type a long line of digits is given — like 4 1 3 5 7 2 9 2 9 8 9 … — and you are asked: “How many 9’s are immediately preceded by 2 and immediately followed by 8?” Every word has a fixed meaning:

WordMeaningExample: 7 5 6
immediately preceded bythe digit right next to it on the left7 precedes 5
immediately followed bythe digit right next to it on the right6 follows 5
betweenboth the before and the after conditions together5 is between 7 and 6

Even when the word “immediately” is not written, the meaning is still “right next to”. If other digits come in between, that digit is not “before” or “after” in this sense.

The window method

  1. From the condition decide which digit must be in the middle (above, 9).
  2. Find it everywhere in the line and look at the digit to its left and right one by one. This is a window of three digits that you slide from left to right.
  3. Mark every window that meets both conditions, and count the marks at the end.
CarefulTwo cares. (1) Overlaps: in 3 3 3 the pair “33” is found twice (the first two and the last two) — count both; and in 2 9 8 9 8 the triple “2 9 8” occurs only once, since the second 9 is immediately preceded by 8. (2) Ends: the first digit has nothing on its left and the last has nothing on its right, so they cannot meet “both sides” conditions.

Conditions may also be given by properties like “even / odd / prime”. Remember: even digits 0, 2, 4, 6, 8; odd 1, 3, 5, 7, 9; prime (single-digit) 2, 3, 5, 7 — note that 2 is both even and prime, and 1 is not prime.

Example 1 · Basic
In the series below, how many 5's are immediately preceded by 6 and immediately followed by 9? 8 5 8 2 3 7 3 6 5 9 4 9 7 7 4 9 9 2 3 8 3 9 7 2
Solution, step by step
  1. 1We look for the triple 6 5 9 in this order.
  2. 2Read the series from left to right and mark every place that fits. Found: 6 5 9 (places 8–10).
  3. 3In all 1 places fit.
Answer 1
In every marked window the middle digit is the one named in the condition.
BasicTry it yourself first
In the series below, how many 5's are immediately preceded by 4 and immediately followed by 2? 8 5 3 4 8 2 5 4 5 2 8 1 3 4 5 2 9 7 3 1 3 4 3 2
BasicTry it yourself first
In the series below, how many times does 7 come immediately after 4? 5 5 7 4 6 2 4 8 4 7 2 2 2 6 6 5 6 5 1 4 7 5 6 4 7 7
BasicTry it yourself first
In the series below, how many 2's are immediately followed by an even digit? 7 8 7 4 6 8 1 5 9 9 5 3 5 7 9 1 3 9 2 2 6 4 2 7
StandardTry it yourself first
In the series below, how many even digits have an odd digit both immediately before and immediately after them? 9 1 3 5 5 3 4 9 6 6 9 9 6 9 9 8 8 2 6 3
StandardTry it yourself first
In the series below, how many digits are equal to the digit immediately before them? 2 6 4 1 2 9 8 3 3 5 8 7 7 3 4 5 2 8 9 4 4 7 9 6
StandardTry it yourself first
In the series below, how many prime digits (2, 3, 5, 7) have an even digit immediately before them? 1 4 4 8 8 8 5 6 1 2 6 8 4 7 6 8 7 3 6 7
StandardTry it yourself first
In the series below, how many digits are such that the digits immediately before and after them add up to 10? 1 5 8 9 2 9 2 1 9 9 4 5 4 6 3 2 2 2 5 1 4 7 1
AdvancedTry it yourself first
In the series below, how many digits are smaller than both of their neighbours? 6 9 9 2 9 5 8 6 4 8 9 1 6 8 9 8 7 5 3 2 1
AdvancedTry it yourself first
In the series below, how many even digits have an odd digit both immediately before and immediately after them? 5 8 4 8 7 1 3 2 6 6 3 5 7 9 3 7 8 1 5 4 4 1 6 8 7 6 9 5
AdvancedTry it yourself first
In the series below, how many 5's are immediately preceded by 6 and immediately followed by 6? 3 9 6 1 3 6 8 8 6 5 6 6 2 7 1 4 8 6 5 6 4 8 5 3 2 5 1 4

Rearranging and altering the digits

Write the original and the new number one below the other

Ascending and descending order, interchanging pairs of digits, reversing a number and adding to or subtracting from digits.

Here something is done to the digits of one number: they are arranged from small to large (ascending) or from large to small (descending), some pairs are interchanged, the whole number is reversed, or something is added to or subtracted from every digit. Then you are asked how many digits did not change place, or which digit stands at a given place in the new number.

OperationHow to write itMind
ascending / descending ordersort the digits into a new linecompare original and new line place by place
interchange first-sixth etc.swap the pairs (i, i + h)write the whole new number, then count
reversing the numberfirst ↔ last, second ↔ second last …with an odd number of digits the middle digit never moves
add / subtractput the new digit under each digitthe results stay within 0 to 9 — the question makes sure of it
Note“Did not change place” means that the same digit stands at the same place — so count places, not digits. If a digit repeats in the number this check matters even more.
Example 2 · Standard
In the number 9441102341 the positions of the first and the 6th digits are interchanged, likewise the second and the 7th, and so on for the following pairs. Which is the 8th digit from the right in the new number?
Solution, step by step
  1. 1There are 10 digits and the pairs are 5 places apart: 1st=9, 2nd=4, 3rd=4, 4th=1, 5th=1, 6th=0, 7th=2, 8th=3, 9th=4, 10th=1.
  2. 2After the interchange the new number is 0234194411.
  3. 3Its 8th digit from the right is 3.
Answer 3
Counting after writing the new number out is far safer than doing it in the head.
BasicTry it yourself first
How many digits keep their position when the digits of the number 785316429 are rearranged in ascending order from left to right?
BasicTry it yourself first
How many digits keep their position when the digits of the number 24619375 are rearranged in descending order from left to right?
BasicTry it yourself first
The digits of the number 71281294 are written in reverse order. How many digits stay in the same position?
StandardTry it yourself first
In the number 71608564 the positions of the first and the 5th digits are interchanged, likewise the second and the 6th, and so on for the following pairs. Which is the 6th digit from the right in the new number?
StandardTry it yourself first
In the number 1547451, 1 is added to every odd digit and 2 is subtracted from every even digit. In the new number, how many different digits appear exactly twice?
StandardTry it yourself first
How many digits keep their position when the digits of the number 64759821 are rearranged in ascending order from left to right?
AdvancedTry it yourself first
In the number 56116387, 1 is added to every odd digit and 2 is subtracted from every even digit. In the new number, how many different digits appear exactly three times?
AdvancedTry it yourself first
In the number 876215 the positions of the first and the 4th digits are interchanged, likewise the second and the 5th, and so on for the following pairs. Which is the 3rd digit from the left in the new number?
AdvancedTry it yourself first
The digits of the number 6791177 are written in reverse order. How many digits stay in the same position?

A group of five three-digit numbers

First the operation on every number, then the order — and tracing back the original

The order of big and small after interchanging or adding-subtracting digits, digit sums and differences.

A group of five three-digit numbers is given and several questions are asked under various conditions. The biggest slip is to make the new order after the operation, when the question was about the original number. So make a small table: the original number on the left, the new number on the right.

  1. Write every number in a row of the table and the new number beside it after the operation.
  2. Put the new numbers in order (as asked).
  3. Read the original form of the number the question is about.
RememberIn order, the bigger number is the one with the bigger hundreds digit; if the hundreds are equal look at the tens, then the units. A bigger sum of digits and a bigger number are different things.
Example 3 · Standard
Numbers: 617, 474, 626, 849, 553. If the first and the third digits of every number are interchanged, which of the original numbers becomes the second largest new number?
Solution, step by step
  1. 1After the interchange: 617→716, 474→474, 626→626, 849→948, 553→355.
  2. 2In descending order: 948, 716, 626, 474, 355. The second largest is 716, which was 617 originally.
Answer 617
Interchanging the hundreds and units digits changes the order completely.
BasicTry it yourself first
Numbers: 514, 955, 398, 482, 841. Which number has the greatest sum of digits?
BasicTry it yourself first
Numbers: 785, 983, 175, 811, 426. What is the difference between the middle (second) digits of the largest and the smallest number?
BasicTry it yourself first
Numbers: 384, 513, 722, 658, 938. If these are written in ascending order, what is the last digit of the middle number?
StandardTry it yourself first
Numbers: 566, 342, 329, 843, 623. If the first and the third digits of every number are interchanged, which of the original numbers becomes the largest new number?
StandardTry it yourself first
Numbers: 178, 627, 417, 187, 249. 3 is added to the first digit and 3 is subtracted from the last digit of every number. Which of the original numbers is now the second smallest?
AdvancedTry it yourself first
Numbers: 342, 117, 162, 363, 283. 3 is added to the first digit and 2 is subtracted from the last digit of every number. Which of the original numbers is now the smallest?
AdvancedTry it yourself first
Numbers: 626, 799, 118, 753, 424. If the first and the third digits of every number are interchanged, which of the original numbers becomes the largest new number?

Forming numbers from digits

Count the choices at each place — first the place that carries the condition

The largest and smallest number, how many numbers can be formed and counting even numbers.

“Which is the largest number from the digits 2, 3, 7?” For the largest, put the biggest digit farthest left (the highest place value): 732. For the smallest put the smallest digit on the left: 237. Their difference is 495.

The way to “how many numbers can be formed?” is choices place by place. To form a three-place number (hundreds, tens, units) from five digits without repeating, there are 5 choices for the hundreds, then 4 left for the tens and 3 left for the units — total = 5 × 4 × 3 = 60.

  • An even number — the units digit must be even; so fill the units first (from the even digits), then the other places.
  • A multiple of five — units 0 or 5; zero cannot be at the hundreds place (the number would no longer have three digits).
  • The place with the condition first — that is the rule, since its choices are the fewest.
Example 4 · Standard
Using the digits 3, 9, 5, 8, 4, without repetition, how many even 3-digit numbers can be formed?
Solution, step by step
  1. 1The number is even only if the units digit is even. The even digits here are 8, 4 — 2 choices for the units place.
  2. 2For the hundreds place any of the 4 left, for the tens place any of the 3 left.
  3. 3Total = 2 × 4 × 3 = 24.
Answer 24
Filling the units first lets the other choices come straight from the digits left.
BasicTry it yourself first
What is the difference between the largest and the smallest 3-digit numbers that can be formed from the digits 1, 2, 4?
BasicTry it yourself first
Using the digits 9, 2, 7, 6, 4, without repeating any digit, how many different 3-digit numbers can be formed?
StandardTry it yourself first
Using the digits 4, 6, 5, 8, without repeating any digit, how many different 3-digit numbers can be formed?
StandardTry it yourself first
Using the digits 8, 5, 4, 3, 7, without repetition, how many even 3-digit numbers can be formed?
AdvancedTry it yourself first
Using the digits 7, 6, 5, 9, 8, without repetition, how many even 3-digit numbers can be formed?
2

Ranking tests

Place in a queue or class, the total, and order from comparisons

Place from the top-bottom and left-right, counting between two persons, changing places, rows and columns, and order from comparisons.

Place and total — four basic formulas

From the top, from the bottom, ahead, behind — one person, two counts

Places from both ends of one person, the total, persons ahead and behind and the formula for one in the exact middle.

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Two persons, changing places and new members in the class

The seats stay, the persons change

Counting between two persons, exchanging seats, new students joining and moving to the middle.

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A class standing in rows and columns

In each direction “sum of both places − 1”, then multiply

Counting the rows, columns and total students of a class standing in a rectangle.

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Order from comparisons — who is more than whom

Height, marks, age, weight — one method for all

Joining the links of comparisons into a full order and reading places, counts and persons between from it.

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3

Time sequence

The clock, days and dates

Reckoning clock times, days of the week, dates and months, working days and the date puzzle, and the order of events.

Adding and subtracting clock times

60 minutes in an hour — not 100

The words for reading time, am/pm, carrying an hour while adding minutes, finding a duration, and fast and slow watches.

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Days of the week — the cycle of seven

The week repeats every 7 days; so the remainder is the real thing

Yesterday-today-tomorrow style days, the remainder on division by 7, day from a date, the days of the months and counting a day in a month.

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Working days and the date puzzle

Write the dates of holidays and subtract; in the puzzle take the common dates

Working days in a month with Sunday and Saturday holidays, and finding the date from the statements of several persons.

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Order of events — who first, who later

Treat “later” as greater and make a chain

Making the full order from comparisons of arriving, leaving or happening.

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4

Strategy guide

The form of the question, the method and the slips to avoid

What to do first in each type, which formula applies when, and where slips happen.

Recognise and apply the method

From the form of the question to the method

A table of the forms of questions and the checklist before every question.

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Common traps and time-savers

Off-by-one slips and their companions

The commonest reasons for losing marks in these questions and how to avoid them.

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5

Mixed practice

Number, ranking and time — first spot the form

Questions on digit strings, places and queues, comparisons, clock, days and dates in three levels.

Basic practice

One formula, one step

Simple counting, the place formulas, straightforward time and day questions.

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Standard practice

Two steps — between, exchanging places, durations, working days

Counting with conditions, questions on two persons and new members, durations, the 1st of the month and working days.

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Advanced practice

Long conditions, six comparisons and mixed questions

Complicated digit conditions, places changing several times, rows and columns, comparison of six persons, hard days and dates and working days.

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6

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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