Browse Category: Tutorials

How to Convert Fractions to Lowest Terms

In the Civil Service Examination and in many mathematics examinations, results that are fractions are usually required to be converted to their lowest terms.  The numerator and the denominator of a fraction in lowest terms cannot be divided by any  similar integer. Knowledge of divisibility rules can be helpful in this process.

Example 1: Convert \frac{6}{9} to lowest terms.

In the first example, we can see that the numerator and the denominator are both divisible by 3. Dividing both the numerator and the denominator by 3 gives us 2/3.

\displaystyle \frac{6 \div 3}{9 \div 3} = \frac{2}{3}

Note that dividing both the numerator and the denominator by the same integer does not change the value of the fraction. Continue Reading

Grammar and Correct Usage Practice 2 Answers and Explanations

Below are the answers and the explanations to the Grammar and Correct Usage Practice Test 2. The incorrect word or phrase in the sentence is highlighted red, while the correct word or phrase is highlighted green.

1. Paul Erdos was a mathematician who writes a lot of research papers in collaboration with other mathematicians.

Correct Sentence: Paul Erdos was a mathematician who wrote a lot of research papers in collaboration with other mathematicians

Explanation: The tense of the verbs in a sentence must be consistent unless there is a reason to change. The verb was is past tense, so the verb writes must be changed to wroteContinue Reading

Fraction Addition Practice Test 1 Solutions and Answers

The idea of getting the least common multiple of the denominator in adding dissimilar fractions is to convert them into similar fractions or fractions whose denominators are the same. Once the fractions are similar, you only need to add the numerator and  copy the denominator.

The solutions to Fraction Addition Practice Test 1 below is divided into three parts: (1) getting the least common multiple of the denominator, (2) converting the given fractions to their equivalent fractions whose denominator is the LCM and (3) adding the converted fractions. Of course, in solving this types of problem the Civil Service Exam, you don’t need to go through all the steps. You should try developing your own short cuts to make solving faster.

Solution and Answers to the Fraction Addition Practice Test 1

Solution to Number 1

Given: \displaystyle \frac{2}{7} + \frac{3}{7}

\displaystyle \frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}

Answer: \displaystyle \frac{5}{7}

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How to Add Fractions

Fractions whose denominators are the same are called similar fractions. Fractions that are not similar are called dissimilar fractions. Hence, the fractions \frac{1}{8}, \frac{3}{8}, and \frac{5}{8} are similar fractions, while the fractions \frac{2}{3} and \frac{1}{2} are dissimilar fractions. In this post, we are going to learn how to add fractions. 

How to Add Similar Fractions

Adding similar fractions is very easy.  In adding similar fractions, you just add the numerator and copy the denominator.  Here are a few examples.

Example 1

\displaystyle \frac{1}{5} + \frac{2}{5} = \frac{1 + 2}{5} = \frac{3}{5}

Example 2

\displaystyle \frac{1}{8} + \frac{2}{8} + \frac{4}{8} = \frac{1 + 2 + 4}{8} = \frac{7}{8} Continue Reading

How to Get the Least Common Multiple of Numbers

In mathematics, a multiple is a product of any number and an integer. The numbers 16, -48 and 72 are multiples of 8 because 8 x 2 = 16, 8 x -3 = -48 and 8 x 9 = 72. Similarly, the first five positive  multiples of 7 are the following:

7, 14, 21, 28, 35.

In this post, we will particularly talk about positive integers and positive multiples.  This is in preparation for the discussions on addition and subtraction of fractions.

We can always find a common multiple given two or more numbers. For example, if we list all the positive multiples of 2 and 3, we have

2, 4, 6, 8, 10, 12, 14, 16, 18, 20


3, 6, 9, 12, 15, 18, 21, 24, 27, 30. Continue Reading

A Gentle Introduction to Fractions

Fractions is one of the mathematics topics that many people have difficulty with. However, unfortunately, it is also one of the most important topics that must be mastered. This is because examination questions in mathematics always include fractions. For example, in the Civil Service Review Numerical Reasoning tests, fractions appear in almost every test: basic arithmetic, number sequences, equations and problem solving.

In this post, we are going to discuss the basics about fractions particularly about the terminologies used. Of course, you don’t really have to memorize them now, but you can refer to this post in the following discussions. In the future discussions, you will use the vocabulary that you have learned here.  Continue Reading

PEMDAS Rules Practice 1 Solutions

Below are the solutions and answers to the problems in PEMDAS Rules Practice 1. Notice that I have color coded the solution to guide you which operation results to which answer. I have also varied the notations like / and ÷ to familiarize you with both of them. In addition, I have also included operations on fractions with expressions in the numerator and the denominator. In a fraction whose numerator and/or denominator contains one or more operations , you have to simplify first both the numerator and the denominator before dividing. The methods in calculating fractions are shown in numbers 7, 9 and 10.

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Subject-Verb Agreement Practice Test 1 Answers

Below are the answers to the Subject-Verb Agreement Practice Test 1. The incorrect verb in each sentence is highlighted red and the correct verb in the corrected sentence is highlighted green. An explanation follows every correction. 

Subject-Verb Agreement Practice Test 1 Answers

1. My brother or my sister are arriving tomorrow.

Correct sentence:  My brother or my sister is arriving tomorrow

Explanation: Two singular subjects connected by or require a singular verb. In this sentence, there brother and sister are both singular, so the sentence should use the singular verb is.

2. Neither Ella nor her friends is available to assist you.  Continue Reading