How to Solve Word Problems by Working Backwards Part 3

In part 1 and part 2 of this series, we have learned how to solve number age problems by working backward. In this post, we are going to learn how to solve backward using inverse operations. Recall that multiplication and division are inverse operations and addition and subtraction are inverse operations.

Example 5

A number is multiplied by 4 and then, 3 is added to the product. The result is 31. What is the number?

Solution

The key phrases in this problem are (1) multiplied by 4 and (2) added to (3) the result is 31. Since we are working backward, we start with 31, and then find the inverse of “added to 3” which is “subtract 3.” So, 31 – 3 = 28.

Next, we find the inverse of “multiplied by 4,” which is “divided by 4.” So, 28/4 = 7.

So, the answer to this problem is 7.

Check: 7(4) + 3 = 31

Example 6

Think of a number. Divide it by 8. Then subtract 4 from the quotient. The result is 5. What is the number?

Solution

The key phrases in this problem are (1) divided by 8 (2) subtract 4 and (3) the result is (3) the result is 5.

We start with the result which is 5 and find the inverse of “subtract 4” which is “add 4.” So, 5 + 4 = 9. Next, we find the inverse of “divide by 8” which is “multiply by 8.” So, 9(8) = 72.

So, the correct answer is 72.

Check: 72/8 – 4 = 9 – 4 = 5.

In the next post, we will discuss more about solving math word problems by working backward.

How to Solve Problems by Working Backwards Part 2

In the previous post, we have learned how to solve number problems by working backward.  In this post, we discuss age problems. We already had 2 examples in the previous post in this series, so we start with the third example.

Example 3

Arvin is 5 years older than Michael.  The sum of their ages is 37.  What are their ages?

Solution

This is very similar to the problems in the previous post in this series.  Arvin is 5 years older than Michael, so if we subtract 5 from Arvin’s age, their ages will be equal.  But if we subtract 5 from Arvin’s age, we also have to subtract 5 from the sum of their ages. That is,

37 – 5 = 32.

Now, their ages are equal, so we can divide the sum by 2.  That is 32 ÷ 2 = 16.  This means that the younger person is 16 since we subtracted 5 from Arvin’s age. Therefore, Michael is 16 and Arvin is 16 + 5 = 21.

Check

16 + 21 = 37.

Example 4

Mia is 3 years older than Pia.  In 4 years, the sum of their ages is 35. What are their ages?

Solution

There are two of them and we added 4 to both ages, so subtracting 8 from 35 (the sum) will determine the sum of their present age.  That is 35 – 8 = 27 is the sum of their present age.

Next, Mia is 3 years older than Pia, so if we subtract 3 from the sum of their present age, their ages will be equal. So, 27 – 3 = 24.

We can now divide the sum of their ages by 2.  That is 24/2 = 12.

This means that 12 is the age of the younger person because we subtracted 3 from the age of the older person.

So, Pia is 12 and Mia is 15.

Check 12 + 15 = 22.

In the next post, we will discuss more problems that can be solved by working backward.

How to Solve Word Problems by Working Backwards Part 1

Most of us would always take a pen and solve for x if we see word problems. But did you know that you can solve them by working backward or even mentally? In this post, I am going to teach you some techniques on solving problems by working backward.

Example 1: One number is three more than the other. Their sum is 45. What are the numbers?

Solution

In the given, one number is 3 more than the other. This means that if you subtract 3 from the larger number they will be equal. Note that if we subtract 3 from one of the numbers, then we should also subtract 3 from their sum. Therefore, their sum will be 45 – 3 = 42. Since the numbers are equal, we now divide the sum by 2. That is, 42/2 = 21.

So, the smaller number is 21 and the larger is 21 + 3 = 24.

Check: 21 + 24 = 45

Example 2: One number is 5 less than the other. Their sum is 43. What is the smaller number?

Solution

This is very similar to Example 1. Here, one number is 5 less than the other; so, if we add 5 to the smaller number, they will be equal. If we add 5 to the smaller number, we should also add 5 to their sum. Therefore, their sum will be 43 + 5 = 48. Since the two numbers are equal, we can divide the sum by 2. That is 48/2 = 24. Since we added 5, it means that 24 is the larger number. So, the smaller number is 24 – 5 = 19.

Check: 19 + 24 = 43

In the next post, we are going to discuss more examples.

Practice Exercises on Subtracting Decimals

We have already learned how to add and subtract numbers with decimals. In this post, we practice subtracting decimals. Recall that in subtracting decimals, the decimal points should be aligned.

Practice Exercises

1.) 2.32 – 1.82
2.) 6.71 – 3.9
3.) 6 – 0.52
4.) 5.03 – 4.25
5.) 0.53 – 0.33
6.) 4 – 1.26
7.) 7.28 – 2.4
8.) 7.08 – 0.29
9.) 3 – 0.305
10.) 40 – 12.5

Answers

1.) 0.5
2.) 2.81
3.) 5.48
4.) 0.78
5.) 0.2
6.) 2.74
7.) 4.88
8.) 6.79
9.) 2.695
10.) 27.5

Enjoy learning!

How to Explore PH Civil Service Exam Reviewer

The PH Civil Service Exam Reviewer website (civilservicereview.com) has already provided free reviewers for the past two years and so far many testimonials have already reached me that this website has been very useful in this review. Below are some of the useful links within the website that are often ignored or missed especially when browsing using mobile phones.

1.) Math – The math page contains basic concepts in numerical ability such as operations on fractions, decimals,  solving equations, as well as finding areas of geometric figures. This also includes topics like LCM, GCF, and addition of positive and negative integers. If you are not very good in math, I suggest that you start with this page.

2.) Word Problems – The Word Problems page contains detailed tutorials about word problems. This includes number problems, age problems, motion problems, work problems, mixture problems, coin problems, investment problems, ratio problems, digit problems, and more.

3.) English – The English page contains reviewers on vocabulary, grammar and correct usage, and paragaraph organization. It also contains basic grammar tutorials particularly the tenses of the verbs.

4.) Practice Tests – This contains practice tests and exercises on both English and Math topics discussed in the site.

5.) Post List – contains the complete of posts of this website. This includes list of Civil Service Exam Passers.

6.) Videos – Useful videos from Sipnayan for learning mathematics.

Another useful site is the Sipnayan Youtube Channel which contains more than 300 Tagalog Math Tutorial videos. 🙂

Lastly, as a bonus, if you already passed the Civil Service Exam, then you may want to explore our partner website, the Philippine Government Jobs, if you want to work for the Philippine Government.

How to Compare Decimal Numbers

We have already learned how to compare fractions and in this post, we are going to learn how to compare decimals. In comparing decimals it is important to understand place value. In the number 213.489, the following are their place values. For the whole numbers,

2 – hundred
1 – tens
3 – ones.

For the decimal numbers,

4 – tenths
8 – hundredths
9 – thousandths

In whole numbers, clearly, the larger the number of digits the larger the number. For example, 821 > 92 since 821 has three digits and 92 has only two digits. Since whole numbers are always greater than decimal numbers in comparing decimal numbers, look at the whole numbers first. Therefore, we have the following rule.

Rule 1: In comparing decimal numbers, look at the whole number first. The decimal numbers containing larger whole numbers have larger values.

Example 1: 84.23 > 82.345 since 84 is greater than 82.

Example 2: 12.56 < 15.001 since 12 is less than 15.

Example 3: 141.85 > 123.4 because 141 is greater than 123

Rule 2: If the whole numbers are equal, then compare the numbers by looking at the tenths place first. The number with the larger digit in tenths place is larger.

Example 1: 18.34 > 18.21 since 3 > 2.

Example 2: 12.95 > 12.15 since 9 > 1.

Example 3: 0. 9 > 0.873 since 9 > 8.

Notice that in Rule 2 Example 3, even if 0.873 has more digits, it is still less than 0.9 since 9 is greater than 8 and they are in the tenths place.

Rule 3: If the whole numbers and the tenths place are equal, then compare first the hundredths place. The number with the larger digits in the hundredths place is the larger number.

Rules 2 and 3 can be generalized. That means that you have to compare the digits from the tenths place first, then hundredths, then thousandths, and so on.

What about negative numbers?

Please take note however the rules of negative numbers.

  • Positive numbers are always greater than negative numbers.
  • If both numbers are negative, do the following:

1.) Make them positive
2.) Apply the rules above
3.) Reverse your answer.

Example: Compare -82.45 and -82.31

1.) Make them positive: 82.45 and 82.31
2.) Apply the rules above:

From the rules above, the whole numbers are both 82, so we look at the tenths place. 0.4 > 0.3, so 82.45 > 82.31

3.) Reverse the answer. Since 82.45 > 82.31, – 82.31 > – 82.45.

7 Tips in Answering Reading Comprehension Questions

Reading comprehension questions is one of the types of test in the Civil Service Examination and is usually included in many Standard English tests. Reading comprehension questions are not very difficult, but they take a lot of time. However, with practice and correct strategy, you might be able to reduce the time you spend on the questions, increase your chance of passing the examination, and even get a high score. Below are some of the tips that you can use in answering reading comprehension questions.

7 Tips in Answering Reading Comprehension Questions

1.) Before reading the passage, read the questions first (not the choices). If you know the questions, then you can choose what to take note while reading.

2.) Pay attention to the first and last sentences in the each paragraph. Usually, those sentences state the main idea of the passage.

3.) Most passages have clues about the important ideas. Phrases like “note that,” “clearly,” and “do not overlook” give you hints on which ideas to focus on.

4.) Read ALL the choices. Do not rush answering if you think you found the correct answer. The next answer might be a better answer.

5.) Take note of the difference between the true answer and correct answer. Carefully read the passage and understand the ideas it communicates. Be sure that your answer is based on the passage and not your own opinion.

6.) Do not spend a lot of time in one question. If you cannot find the answer even if you refer to the passage, eliminate the obviously wrong choices, and choose your answer from the remaining choices.

7.) Review your answers. If you finished the exam early, go back to the questions where you have doubts. Reread the passage and answer the question.

I hope these tips can help you in your endeavor to pass the Civil Service Exam or any other exam for that matter. Good luck and share these tips to your friends.

Solving Quadratic Word Problems in Algebra

Quadratic Equations are equations of the form ax^2 + bx + c = 0 where a, b and c are real numbers and a \neq 0. Depending on the form of the equation, you can solve for x by extracting the quare root, factoring, or using the quadratic formula.This type of equation appears in various problems that involves multiplication and usually appears in the Civil Service Exams.

The following series details the method and strategies in solving problems involving quadratic equations.

How to Solve Quadratic Word Problems Part 1 is about solving problems involving consecutive integers. In this problem, the product of consecutive numbers is given and factoring was used to solve the problem.  » Read more

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