The sum of two numbers is 78. One of these numbers is 6 more than the other. Of the two numbers, which is the smallest?

This is a classic example of an ASVAB Arithmetic Reasoning problem, where you are supplied with the data in written form and must deduce the correct answer from the information provided. How did you manage to solve this problem? Did you think of a way to overcome it in a simple and effective way? The best method in approaching this problem would be to employ algebra. You could, of course, work out the problem by guessing the two numbers but this is a long and laborious task, and in a short exam, you simply don’t have the time!

The correct answer to this problem is 36, and we’re going to use algebra to discover it. Let one of the numbers you don’t know be x and let (x + 6) represent the larger number. By forming a simple equation using these two terms, we can work out the value of x. When we have this value, we can determine the larger number by returning to (x + 6). Let’s take a look at how to solve the problem:

Given that (x + 6) represents the larger number, we can substitute 36 back into x, in which case we determine the larger number to be 42. As it happens, 36 and 42 total 78, so we know we have determined the correct answer. However, the question specifically requested we calculate the smaller of the two numbers, so we accept 36 as the correct answer. As we stated previously, you don’t necessarily need algebra to solve this problem, but it makes it a whole lot easier and quicker to solve.

A waiter earns approximately 11% tips on the total value of the food he serves during the night. If the waiter serves $470 food on Monday night as well as $240 on drinks, what total value of tips has he earned that night?

This is a standard percentage question that regularly appears on the ASVAB Arithmetic Reasoning exam. This particular example is testing two things: your ability to successfully read the aims of the question, while asking you to correctly use percentages in a real-life scenario. Many of you probably rushed through this question and tallied the total value of food and drinks sold on Monday night – totaling a handsome $710. However, it would be wrong to use this value to calculate the tips involved. This is because the question informed us that tips can only be calculated from the total value of food sold during the night – it mentions nothing of the value of drinks.

This means we can discard the drink total and, instead, focus purely on the value of food – $470. The waiter will earn 11% tips on the value of this food, so all we need to do is find the value of 11% of $470. The long form would involve creating the following fractions:

If we divide 5,170 by 100, we determine the answer to be $51.70 – which is the correct answer. If you are not comfortable with the long method, you could always choose the short method, which simply involves multiplying the total value of the food by 0.11. After all, 0.11 is the decimal form of 11%.

It’s useful to bear multiple methods in mind, as it expands your mathematical skill-set in solving these problems. In addition, knowing multiple methods allows you to verify if you calculated the correct answer, as you’ll be able to perform both methods and see if the result came out the same. Either way, the second method, using the decimal form, is generally considered easier and quicker, as long as you form the correct decimal number.

In this particular problem, we need to possess knowledge of unit conversion and area. The problem itself isn’t particularly taxing when you know how to convert from one unit to another. The first step we need to take is to determine the size of the room – which works out at 308ft2, as all we need to do is multiply 14ft by 22ft. With the total area of the room now within our grasp, we now need to find out how much varnish we require and the cost therein. However, we have a problem. The varnish is priced per square yard, even though our room is measured per square foot.

To convert square feet to square yard, we only need to divide by 9. This is by virtue of the fact that 1 square yard is the same value as 9 square feet; take a look:1yd^2 = 3ft x 3ft = 9ft^2

We can see that 1 yard is equal to 3ft, so 1 square yard is the same as 3ft x 3ft, which works out as 9 square feet. When we divide 308ft2 by 9, the value works out as 34.22yd2. With the area of the floor now measured in square yards, all we need to do is multiply the cost of the varnish per square yard by the area of the floor.

So, it will cost $15.40 to pay for the varnish for the entire area of the floor in question. You’ll need to become familiar with unit conversions such as this, as well as being comfortable dealing with calculations involving area, perimeter, diameter, scale, and proportion.

In this type of question, you are given the value of x. Knowing this, your first step should always be to insert 6 into wherever you find the value x. When we complete this step, all we need to do is tidy up both sides of the equation, isolate y, and find the final value.

If you are new to algebra, you should probably only try one step per line. This is the method chosen in the example above. With time, though, you will develop the ability to do multiple steps at the same time, meaning the equation becomes simpler, quicker, and easier to solve. We would advise against rushing to this stage, as it’s simply not worth the extra potential mistakes. We do advise, however, that you learn the basic structural steps in solving this type of problem in the first place, steps such as:

We solved the brackets first before turning to any other part of the equation. This is due to the BOMDAS Rule which states you order how you answer questions in the following way: Brackets, of Multiplication, Division, Addition, Subtraction. Always follow the BOMDAS Rule if you’re unsure how to start and progress through an algebra problem.

Notice how we solve one side of the = side at a time. This eliminates confusion about what’s going on, while isolating what we require (the value of y) on its own on the left-hand side.

This allows us to eliminate the 3y into y as we divide the other side of the equation by 3. In essence this last step involves dividing both sides of the equation by 3 in order to eliminate it from the equation. After all, 3 divides into 3 to leave 1, meaning we’re left with 1y which is the same as y.

By following the logical steps of algebra, with particular emphasis on using the BOMBAS Rule, we can easily arrive at the correct answer.

Martin scores the following results in four of his ASVAB tests: 73, 77, 63 and 71. What score will Martin need to achieve in his fifth ASVAB test to score an average of 70 for all five tests?

There are two ways of working out this problem. First, though, you need to understand the concept of what it means to find the average of numbers. The average of 6 and 8, for example, is 7, meaning we added up the total (6 + 8) and divided by the amount of numbers we have (2). In this ASVAB Arithmetic Reasoning question, it’s asking us to calculate what score Martin will need to achieve in his next exam to score an average of 70 for all five exams he would have taken. So for this question, we could do two things. We could manually check the answers and divide them by 5, but this would be a time consuming process – as we are limited by the time restrictions imposed by the exam.

The quickest and easiest way, again, refers to using algebra. You should think of algebra as a friendly tool, something you can call upon in times such as this. Using algebra is quick, effective, and most importantly, makes an awful lot of sense when you get used to it. Keep practicing the use of algebra and you’re sure to benefit in the long run. The question is – how are we to use algebra in a question such as this? Well, let’s take a look at the following equation we could easily set up to solve the problem once and for all:

This is a mathematical reflection of the problem described in written form above. We can see how the five required numbers are located on top; the desired average of 70 is also depicted, as is the method of finding averages; through dividing by 5 in this particular case. Let’s complete this equation by first totaling the top line of average values:

So, in other words, Martin will need to score 66 in his next exam to manage an average score of 70 over all five of his ASVAB tests. You can see how algebra, yet again, has come to save the day. Using x to determine ‘what you are looking to find’, as in this example and the example on numbers above, prove very useful tools you can apply across a myriad of different question types. You should study these two examples again to ensure you fully grasp how to employ algebra over the course of your ASVAB Arithmetic Reasoning test.