Entrance exams: calculating and fractions

Calculating that carries marks in entrance exams

The next four lessons are about entrance exams. Before we start, one thing: entrance exams are not a genius test. They are a test of calm, reliable work. And that can be trained — that is exactly what we will do now.

Today the first pillar: calculating with numbers, fractions and decimals. These questions open the test and they are the surest marks that are there. You will not learn anything new — you already know it all from earlier. We train reliability: finish a calculation without a needless mistake.

The recipe for reliability has three parts. Write the method — even middle steps you “see in your head”. A head under pressure makes mistakes, paper does not. Watch signs and decimal points — most lost marks are copying and signs, not not knowing. Check backwards — substituting or a rough estimate spots a result that has run away.

Mistakes in training are good news: each one you make now is one you will not make in the test. Let’s go.

Tereza at home ran a practice test for the first time and is sad about the result: “Half the mistakes are from stupidity. Here a minus, here a decimal point one place over…” Her older brother, who passed entrance exams last year, smiles: “Welcome to the club. I had it the same at the start. Then I made a rule: every step on paper, even the smallest. In two weeks those needless mistakes almost disappeared.” Tereza tries another test with a paper method. Half as many mistakes. She is not silly — she just used to count in the air.

Typical questions and how to do them

Type 1: Order of operations. A question like 18 − 2 · (3 + 4). First the bracket (7), then multiplying (14), then subtracting (4). The rule: brackets → multiplying and dividing → adding and subtracting. Write one step per line.

Type 2: Fractions. Adding through a common denominator, multiplying straight, dividing by the reciprocal fraction. Always cancel the result — an uncancelled fraction can cost a mark.

Type 3: Decimals. Adding and subtracting under each other, decimal point under decimal point. Multiplying by ten and a hundred = shifting the decimal point. Watch the places: 0.3 · 0.2 = 0.06, not 0.6.

Type 4: Negative numbers. Minus times minus is plus. Subtracting a negative number is adding: 5 − (−3) = 8.

Type 5: Conversions. Fraction → decimal (divide), percents → hundredths (divided by a hundred). Be able to do both ways.

A check ritual at the end of every question (10 seconds): 1) Estimate — is the result reasonably big? 2) Last step — did I copy the number right? 3) The question — am I answering what they asked? This ritual is the most profitable ten seconds of the whole test.

Example 1: order of operations under pressure

Question (a test type): Calculate 36 − 4 · (2 + 5) + 8 : 2.

Method line by line:

Bracket: 2 + 5 = 7. Multiplying: 4 · 7 = 28. Dividing: 8 : 2 = 4. Left: 36 − 28 + 4. From the left: 36 − 28 = 8, then 8 + 4 = 12.

Answer: 12.

A common mistake: calculate 36 − 28 + 4 as 36 − 32 = 4 — as if plus belonged in the subtracting. Adding and subtracting go left to right, no combining first.

Example 2: fractions in one expression

Question: Calculate 3/4 + 1/6 and write the result in simplest form.

Method: A common denominator of four and six: 12.

3/4 = 9/12 (expanded by three). 1/6 = 2/12 (expanded by two).

Sum: 9/12 + 2/12 = 11/12.

Check: 11 and 12 have no common factor — the fraction is in simplest form.

Estimate: 3/4 is 0.75 and 1/6 about 0.17, the sum around 0.92. And 11/12 ≈ 0.92. It matches.

Answer: 11/12.

Example 3: decimals and the decimal point

Question: Calculate 3.6 · 0.25.

Method: I calculate without decimal points: 36 · 25 = 900.

Now the decimal points: 3.6 has one decimal place, 0.25 two — together three. I shift the decimal point three places: 0.900 = 0.9.

Estimate check: 0.25 is a quarter. A quarter of 3.6 is 0.9. It matches exactly.

Answer: 0.9. The quarter trick shows a second way: 0.25 = 1/4, so times 0.25 = divided by 4. Converting to a fraction sometimes calculates faster.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

A mix like in entrance exams. Each question different — equations, percents, fractions, geometry. Write fractions as a/b, decimals with a comma.

Practise the questions

On paper

A practice set. Time yourself, but do not rush at the cost of the method — now we train accuracy, speed will come on its own.

1. Calculate with a method line by line: 48 − 6 · (3 + 4), then 2.4 + 0.36 · 10, then 7 − (−5) + (−3).

2. Fractions: 2/3 + 1/4, then 5/6 − 1/2, then (2/5) · (3/4). All in simplest form.

3. Conversions: write 3/8 as a decimal, 0.45 as a fraction in simplest form and 12 % as a decimal.

4. For every question do the check ritual: estimate, copy, the question. Write down how many mistakes the ritual spotted.

Now you 💪

1. You write every step on paper, even the one “visible in your head”.
2. You hand in fractions in simplest form and with decimals you count decimal places.
3. At the end of every question you run the check ritual: estimate, copy, the question.

What to take from this lesson

• Entrance exams test reliability, not genius. Reliability is trained.
• Every step on paper. Counting in the air does not pay.
• Order of operations: brackets, then times and divide, then plus and minus from the left.
• Do not mix fractions with decimals — convert to one.
• The check ritual (estimate, copy, the question) costs 10 seconds and carries marks.

© 2026 Ing. Martin Polak / AlgoRhino · 内容使用条款