How to Solve Digit Problems Part III

This is the third part of the tutorial series on Solving Digit Problems. In Part 1 and Part 2, we used one variable to solve digit problems. In this post, we learn how to use two variables to solve digit problems. We still use the problem in Part 2.


The sum of the digits of a 2-digit number is 9. If the digits are reversed, the new number is 45 more than the original number. Find the numbers.

Solution and Explanation

Let t = tens digit and u = units digit.

From the first sentence in the problem, we know that

t + u = 9 (1).

Also, as we have learned in the first two parts of this series, 2-digit numbers with tens digit t and ones digit u can be represented (or has value) 10t + u. For example, the number 25 with t = 2 and u = 5 has value 10(2) + 5.

So, the we can represent the original number as

10t + u.

If we reverse the digit, the specific example which is 25 becomes 52. This becomes 10(5) + 2. Hence, we can represent the reverse number as

10u + t.

Therefore, we our representation is as follows:

original number: 10t + u
new number (with digits reversed): 10u + t.

In the second sentence in the problem, it says when the digits are reversed, the new number is 45 more than the original number. That means that if we add 45 to the original number, they will be equal. That is,

original number + 45 = new number. 

Substituting the representations above, we have

10t + u + 45 = 10u + t.

We can simplify the equation by putting the variables on the right.

45 = 10u + t – (10t + u)
45 = 10u + t – 10t – u
45 = 9u – 9t (2).

Thus, we have 2 systems of equations.

t + u = 9 (1)
9u – 9t = 45 (2).

Note: We just change reverse the position of the expressions in equation (2).

We can solve this using elimination or substitution. In this solution, we use substitution.

First, we find the value of t in (1)

t + u = 9.

Subtracting u from both sides, we have
t = 9 – u.

Next, we substitute 9 – u to the value of t in (2)

9u – 9t = 45
9u – 9(9-u) = 45
9u – 81 + 9u = 45
18u – 81 = 45
18u = 45 + 81
18u = 126.

Dividing both sides by 18,

u = 7

So, the units digit is 7.

To find t, we substitute in one of the equations in (1) and (2). We substitute in (1),

t + u = 9
t + 7 = 9
t = 2

So, our number has tens digit 2 and ones digit 7. Therefore, the number is 27.

If we check, if the number is reversed, it becomes 72. Let’s see if the number with reversed digit is 45 more than the original number.

72 – 27 = 45.

Therefore, we are correct.

Having two equations in two variables is an example of systems of equation. In the process above, we solved for the value of one of the variables in (1) and substituted it in (2). We will discuss systems of equations, particularly linear equations in two variables in details in the next posts.

3 Mistakes Done Before or During the Civil Service Exam Day

During the Civil Service Exam day, it is important to be prepared, but what is more important is not to get late. The most important goal during the day of the examination is to arrive at the venue ahead of time. In this article, we learn from mistakes done by previous examinees. Please keep them in mind.

1. They sleep very late.

Some examinees usually sleep very late the night before the Civil Service Examination. One potential drawback about this is waking up late. There was one examinee in May 2015 who woke up 9:00 am. The start of the examination was usually 8:00 am.

civil service exam day

image via Wikipedia

In any examination, it is always advisable to relax the night before. This will make you fresh during the day of the examination. Sleepiness during exams can make you slower and can affect your focus.

2. They wake up very early.

There are some cases in Civil Service Examinations where examinees sleep early but also wake up very early. This is probably due to excitement (wink) or nervousness or being afraid to be late. However, many of them experienced headache during the day of the examination making it difficult for them to answer the questions.

You can set your alarm clock during the day of the examination. Better still, set two alarm clocks. I usually do this if I have some important appointments. If you wake up too early, at least you can sleep a bit more without being worried.

And it is not advisable to review if you wake up early!

3. They don’t search ahead for the venue.

Many examinees only search for the venue during the day of the examination. Many of them get lost and some even get late. Note that the examiners don’t allow late comers during the exam!

Related: 5 Things to Do a Day Before the Exam

It is important to know the venue of the Civil Service Examination ahead of time. It is even better to visit the venue before the examination and know some alternative routes as well as traffic conditions. To those who are very far the venue, you can sleep at a friend’s house or even rent a hostel.

And most of all, during the day of the examination, don’t forget your pencil. Simple things count!

Idiomatic expressions – Line

Written by Nigina Dustova

This is the third part of our series on idiomatic expressions.  The first part is about idioms containing the word back and the second part is about idioms containing the word ground. In this post, we are going to learn about some idioms about line.

Line is a quite a common word in many idiomatic expressions. It has a range of
If you drop someone a line, you write them a letter.
The bottom line refers to accounts where you see the profit or loss at the bottom.
The dotted line is the line on which you sign your name for a contract or bill.
If you draw a line under something on a piece of paper you separate it from what comes before. If you draw a line under an experience, it means it is finished and you don’t want to hear any more about it.
If you toe the line, you conform. In politics you often have to toe the party
line – support what the party believes. People who rock the boat are said to step
out of line. If something is different from other similar things, it can be brought
into line with the others.
If your job is on the line, you are at risk of losing it. If someone is on the
line, they are on the telephone.


Susan says she’s happy with Martin but reading between the lines, something’s
not quite right. They never seem to go out together.

This is Sarah Williams. Can I speak to Amanda Price, please?
Hold the line, please. I’ll just see if she’s free.

Right. Have a good journey. Good luck with the new job and everything.
Thanks. I’ll drop you a line to let you know how I’m getting on.

I agree. Let’s move the meeting to the 24th and involve the other departments.
Good. I’m pleased we’re thinking along similar lines.

Exercises: Use the following idiomatic expressions in the sentences below:
a. in line for
b. the bottom line
c. on the dotted line
d. drew a line under
e. out of line

1. Make sure you read the contract in detail before you sign __________.

2. If you step ________ in this company, they get rid of you. You don’t get a second chance.

3. When John’s divorce came through, he ________ that episode of his life.

4. Helen’s doing very well at work. In fact, I think she’s __________ promotion.

5. I know you don’t want to learn English but ___________ is, if you want to
get a better job here, you need to speak a foreign language.

c; 2. e; 3. d; 4. a; 5. b;

Idiomatic expressions – Ground

Written by Nigina Dustova

The ground is one of the most basic things we know. It is not surprising then if ground rules are basic rules. In a similar way, if you have a good grounding in a subject, you know the basics.
Hundreds of years ago, everyone had to hunt animals to stay alive. If you wanted to know if there were any large animals around, you put your ear to the ground and listened. Today if you keep your ear to the ground you are trying to be aware of any ideas or rumors which are around.
Flying is a more modern idea. A plane literally gets off the ground. This can be applied to a new project in business.


1. Paul’s good with people; he’s patient; he’s organized. Personally, I think social
work will suit him down to the ground.
2. Another boring meeting! They’re a waste of time. We always cover the same old ground.
3. I knew I hadn’t stolen anything in the shop, so I just stood my ground and eventually they believed me. They had mixed me up with someone else!
4. The best advice my grandmother ever gave me was to keep my feet on the ground even if my head was in the clouds!

Exercises: Use these idiomatic expressions in the sentences below:
a. ground rules
b. on dangerous ground
c. a bit thin on the ground
d. keep your ear to the ground
e. a good grounding
1. – Excuse me, I’m thinking of changing my PC for an Apple Mac. Have you got any
second-hand ones?
Sorry, we haven’t. Second-hand Macs are __________ at the moment.

2. – We’ve got to do something to reduce our costs.
– Well, if you start talking about cutting staff, you’re ___________.

3. – Thanks for letting me stay with you. You’re doing me a big favor.
– That’s OK, but there are one or two ___________ : no smoking, shoes off at the front door and everyone does their own washing up.

4. – You’re doing a computer course, aren’t you?
– Yes, it’s nothing special but it gives you __________ in the basics.

5. You want to move to the Personnel Department, don’t you? I’ve heard there might
be a job there soon so ____________.

c; 2. b; 3. a; 4. e; 5. d;

How to Write Order of Adjectives

Written by Nigina Dustova

We can use more than one adjective before a noun. The order of adjectives in front of a noun is as follows:

1.) quality / opinion + size, etc. The most general adjective usually comes first:
a beautiful large French handmade teak cupboard

2.) size / age / shape / color + where from? + past participle:
a large French handmade cupboard

3.) Where from? + past participle:
a French handmade kitchen cupboard

Or: past participle + where from?
a handmade French kitchen cupboard

We order adjectives according to their meaning. This is the normal order:
1. Opinion (how good?)
2. Size (how big?)
3. Age (how old?)
5. Color (which?)
6. Origin (where from?)
7. Material (made of?)
8. Type (what kind?)
9. Purpose (what for?)

a small green insect (size, color)
Japanese industrial designers (origin, type)
a wonderful new face cream (opinion, age, purpose)
a long boring train journey (size, quality, type)

Exercises: Put the adjectives in proper order. Begin each sentence with I’m looking for…

  1. clock radio – white – Taiwanese – cheap – for my bedside table
  2. polished – beautiful – antique – dining-table – mahogany – English
  3. canvas – American – a pair of – trainers – grey and red – which I can use for jogging
  4. cottage –stone-built – small – old – country
  5. cotton – dress – summer – pink and white – for my holiday


  1. a cheap white Taiwanese clock radio for my bedside table
  2. a beautiful antique English polished mahogany dining-table
  3. a pair of grey and red American canvas trainers which I can use for jogging
  4. a small old stone-built country cottage
  5. a pink and white cotton summer dress for my holiday

Idiomatic expressions – Back

Written by Nigina Dustova

An idiomatic expression is a phrase or a fixed expression that has a figurative, or sometimes literal, meaning. An idiomatic expression’s figurative meaning is different from the literal meaning. In this post, we are going to learn idiomatic expressions which use the word back.


Your back is often associated with hard work. If you put your back into something, you work very hard at it.
If you pat someone on the back, you are congratulating them.
If someone is on your back all the time, they may be checking up on you.
You want them to get off your back and leave you alone.
If you have your back to the wall, literally, you are about to be killed by a firing
squad. Metaphorically, it means you are in a desperate situation.
If you want to say that someone knows very little about something, you can say:
What James knows about finance could be written on the back of a postage stamp.


1. – Listen Ben, when are you going to get out of the house and start looking for a job?
– I’m trying my best. Now would you get off my back and give me a break!
2. – I rang the police when I saw something strange going on across the road and they caught a couple of burglars red-handed!
– You certainly deserve a pat on the back!
3. If you want to criticize me, tell me to my face. If there’s one thing I hate it’s people who talk behind my back.
4. – I’ll never be able to clear the garden by Friday. It’s impossible.
– You will if you put your back into it. You don’t know what hard work is.

Exercises: Use the following idiomatic expressions with back in the sentences below:
a. backs to the wall
b. in the back of beyond
c. like the back of my hand
d. like water off a duck’s back
e. take a back seat
1. My parents moved away from the city when they retired. Now they live ________, about two hours drive away.
2. If I want to pass my exams, I think my boyfriend will have to __________ for a few weeks.
3. I get criticized a lot at work but it’s part of the job. I’m used to it now. It’s _________.
4. A lot of export companies have their __________ at the moment. Interest rates must come down even further if things are to get any better.
5. I’ve been to Dublin lots of times. I know the place _____________.

1. b; miles from anywhere. 2. e; be less important. 3. d; it doesn’t affect me. 4. a; in a very difficult situation 5. c; in a very difficult situation

A Summary of the Perfect Progressive Tenses

Written by Leny Ortega

Now that we are done with the discussion on the perfect progressive tenses: past perfect progressive, present perfect progressive, and future perfect progressive. Below are practice exercises to test your mastery of the previous lesson. Answers are provided after the exercises for easy self-checking.

Write the correct form of the perfect progressive tense used in each given sentences.

1. Senate probe on Maguindanao clash (go) on for several months now.

2. I (clean) the kitchen when my friend (arrive).

3. She (receive) negative comments from social media since last week.

4. A few weeks from now, Korean singers (perform) in front of many Filipino fans.

5. When I (arrive), the principal (talk) to students about the incident.

6. George (edit) his speech when the emcee (tell) him he is the next speaker.

7. Jeanne (tell) her friends that the project is long over due since last week.

8. By next week, the doctor (check) the condition of the pregnant woman.

9. A few weeks from now, she (reveal) her real health condition.

10. Annsly (apply) for a job since last month.

Answer key

1. Has been going

2. had cleaning, arrived

3. Has been receiving

4.will have been performing

5. Arrived, had been talking

6. Had been editing, told

7. Has been telling

8. Will have been checking

9. Will have been revealing

10. Has been applying

Grammar Tutorial: Future Perfect Progressive Tense

Future perfect progressive is used to state an action that is perceived to be continuously happening when a certain time comes.

Form: will have been + present participle

1. By this afternoon, the teachers will have been arriving to attend the meeting.
2. Few months from now, politicians will have been preparing for the forthcoming national election.

Write the correct future perfect progressive form of the given verb inside the parentheses.

1. By tomorrow, the patient (recover) from the trauma.

2. Four years from now, my friend (complete) her papers to Australia.

3. By the next decades, scientists (explore) different planets.

4. A few weeks from now, teachers (train) for the K12 seminar for this coming school year.

5. By tomorrow, Myrna (visit) her sick mother for a long time.

1. Will have been recovering
This means that before tomorrow ends the patient already recovered from trauma.

2. will have been completing
The idea here is that before the four Years of preparation ends my friend already completed all papers document for Australia.

3. Will have been exploring
This sentence tells us that in the coming decades scientists will try exploring different planets.

4. will have been training

In this particular sentence, it gives us a picture of the teachers in the k12 seminar which will last for a certain week starting today.

5. Will have been visiting
This means that before the day tomorrow ends Myrna already came to visit her ailing mother for a long time.

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