GMAT Math Made Easy

We have been learning quite a bit about how to manipulate fractions. In this post, we will study how to set up equations for word problems. These problems are known to be quite tricky among many GMAT takers. A simple trick to do better on the word problems is to remember the following sentence: Don't forget your X . We explain this by studying the following example. Example . A company is giving a 10% discount for every product it sells. After this discount, a product was sold at 10 USD. How much was the product prior to this discount? To approach this, you need to set $X$ to be something you want, namely price of the product before the 10% discount. In math $1$ means the ``whole'', and this is the same as 100%. If we take out 10%, what we are left is 90%, and this is the same as $0.9$. This means that after 10% discount, what we get is 90% of the original, which means $0.9$ of the original, which is $(0.9)X$ in this case. The problem says that this is equal to $10$ (in

One of many questions GMAT can ask you is how to compare two or more fractions (i.e., which fractions are bigger/smaller). We will discuss a very simple method for comparing fractions. How to compare two fractions . To compare two fractions, we need two steps. Step 1 . Make the same positive bottoms. For example, let's compare $1/13$ and $2/25$. To make the same bottom, you multiply the opposite bottom up and down. For $1/13$, the opposite bottom is $25$, we have $$\frac{1}{13} = \frac{1 \cdot 25}{13 \cdot 25} = \frac{25}{13 \cdot 25}.$$ Why didn't we compute $13 \cdot 25$? Just wait for it now. We will explain why after comparing $1/13$ and $2/25$. For $2/25$, the opposite bottom is $13$, so $$\frac{2}{25} = \frac{13 \cdot 2}{13 \cdot 25} = \frac{26}{13 \cdot 25}.$$ Step 2 . Compare the tops. We continue the example from Step 1. We have $$\frac{1}{13} = \frac{25}{13 \cdot 25} < \frac{26}{13 \cdot 25} = \frac{2}{25}.$$ This implies that $1/13$ is smaller than

So far, we have been studying some essential properties of fractions. In this lesson, we will learn how to multiply two fractions. How to multiply fractions . Multiplying two fractions is much simpler than adding them. That is, you multiply tops with tops and bottoms with bottoms: $$\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}.$$ Exercise . Compute $$\frac{2}{100} \cdot \frac{5}{100}.$$ Solution . We have $$\frac{2}{100} \cdot \frac{5}{100} = \frac{2 \cdot 5}{100 \cdot 100} = \frac{10}{10000} = \frac{1}{1000}.$$ Note that the last equal sign can be checked by reviewing  this previous lesson . Don't be confused between addition and multiplication . We have $$\frac{2}{100} + \frac{5}{100} = \frac{2 + 5}{100} = \frac{7}{100} = \frac{70}{1000},$$ while $$\frac{2}{100} \cdot \frac{5}{100} = \frac{2 \cdot 5}{100 \cdot 100} = \frac{10}{10000} = \frac{1}{1000},$$ so they are not equal. The first is bigger than the second (because they have the same bottom). Ide

We have been studying how to work with fractions. In particular, we have learned how to add and subtract with fractions: Lesson 1 .  Adding fractions . Lesson 2 .  Subtracting fractions . How to check whether two fractions are equal . Recall that one fraction can be written in various ways (e.g., $1/2 = 2/4$). In this posting, we study how we can identify various different looking fractions are the same. For example, we have $$\frac{2}{3} = \frac{10}{15}.$$ How do we know? One way to check is to multiply the opposite bottoms up and down. That is, we have $$\frac{2}{3} = \frac{2 \cdot 15}{3 \cdot 15} = \frac{30}{45}.$$ We also do this for the other fraction $10/15$: $$\frac{10}{15} = \frac{10 \cdot 3}{15 \cdot 3} = \frac{30}{45}.$$ Therefore, we have $$\frac{2}{3} = \frac{30}{45} = \frac{10}{15},$$ so ignoring the middle we must have $2/3 = 10/15$. The key here is to multiply opposite bottoms up and down to get the same bottom ! Now, notice that because we get

In the previous lesson , we have learned how to add fractions together. Let's first connect this to a formula you may have seen in high school. We have $$\frac{A}{B} + \frac{C}{D} = \frac{AD}{BD} + \frac{BC}{BD} = \frac{AD + BC}{BD}.$$ That is, we have first multiplied the opposite bottoms ($D$ is the opposite bottom for $A/B$, and $B$ is the opposite bottom for $C/D$) up and down and then keep the same bottom $BD$ and added the two different tops. In many curricula, teachers describe this process as follows. Multiply the bottoms. Cross multiply: $A$ with $D$ and $C$ with $B$. Then add the resulting fractions. The method introduced in the  previous lesson  is equivalent to this cross multiplication method, and in fact, it teaches you why this cross multiplication is valid. You may use either way, but you will have a stronger understanding if you know both. (If you can forget one of them, you can use another!) Subtraction of fractions . We can subtract a fraction from

The notion of fractions  is a fundamental concept in math. As soon as we learn how to count $1, 2, 3, \dots$, the next natural step is to understand the notion of proportion. Generally, speaking, the following three words have the same meaning: fraction, proportion, ratio. More precisely, proportion/ratio of $A$ to $B$ means $A/B = \frac{A}{B}$. Exercise . What is the ratio of $2$ to $4$? Solution . Ratio of $2$ to $4$ is $2/4$, so the answer is $2/4$. Perhaps, some people would argue that the answer to the above exercise is $1/2$. That's also true. In fact, we have $$\frac{1}{2} = \frac{2}{4}.$$ Think about it. Having one piece out of two equal-size pieces of a cake is the same as having two pieces out of four equal-size pieces of the cake. This means that there are multiple ways to write a single fraction. The best way to see this is the trick of multiplying $1$ . What do I mean by this? Step 1 . Note that multiplying $1$ does not change anything: $$\frac{1}{