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 USD), so
$$(0.9)X = 10.$$
Don't forget your $X$! We are trying to solve for $X$. How do we manipulate this to get $X$? The easiest way is to multiply $10$ on both sides. Then we get $9X = 100$. Then we divide both sides by $9$ to get $X = 100/9$ (USD). If you like decimal expressions, the answer would be $X = 11.111111111...$ (USD), but there is no way we can stop this expression, so we either have to keep $...$ or say $100/9$ USD is the answer.
Remark. Let me provide another example, but it turns out to be slightly harder than I wished. However, this example is certainly within the range of GMAT's difficulty, so take a deep breath and study it when you have enough energy!
Example (Hard). Store A and B sell apples. Originally, the price of one apple from either store was 1 USD. However, the stores started to compete each other by starting their discount programs. Store A gives you 10% discount when you buy 10 apples. On the other hand, Store B gives you one more apple when you buy 10 apples. If we compare the price per one apple which store sells apples cheaper if we purchase 10 apples (meaning 11 total apples for Store B)?
Remark. I really like this problem, because when I first tried it, I thought both stores sold the apples at the same price. As you will see, it turns out they didn't, and I think it captures a great strategy among marketers: a customer can feel that they are purchasing at a cheaper price even if they don't. This strategy is used in many places, and one instance where I experience this is Starbucks Rewards program.
How do we start? Don't forget your X! You first need to set $X$ to be something you want to find. First, we want to find the price of one apple Store A sells when we buy 10 apples from them, so we let
$$X = \text{ price of one apple when we buy 10 apples from Store A}.$$
How do we compute this? This is where much thinking comes in. It is difficult to find $X$ right away, but it is easier to find $10X$ because that is the price of 10 apples from Store A. Originally this was supposed to be 10 USD, but under their discount program, we get 10% discount. We need to recall that 10% means $1/10$ because 100% means $1$. Since Store A cuts out $1/10$ of the price, what we need to pay for 10 apples is $9/10$ of 10 USD. Thus, we have
$$10X = \frac{9}{10} \cdot 10,$$
where the unit is given by USD, but we chose to suppress it. The right-hand side can be manipulated as
$$\frac{9}{10} \cdot 10 = \frac{9}{10} \cdot \frac{10}{1} = \frac{9 \cdot 10}{1 \cdot 10} = \frac{9}{1} = 9.$$
Combining this with the previous equation, we have $10X = 9$, so dividing this by $10$ both sides, we have $X = 9/10$. Don't forget what your $X$ meant: it meant the price of one apple if you buy 10 apples from Store A. The conclusion so far is: the price of one apple if we buy 10 apples from from Store A is $9/10$ USD.
Now, we need to do the same procedure for Store B. Your $X$ needs to be what you want to find, so we set
$$X = \text{ price of one apple when we buy 11 apples from Store B}.$$
First, carefully note that now we work with 11 apples rather than 10 apples because this is what problem said in the parenthetical remark. How do we find this X? You need to think quite a bit and read the problem over and over until you have some sense of where to start. One way to start is to noting that it is easier to find the price of 11 (total) apples from Store B. Originally it is supposed to be 11 USD, but one of them comes free, so we pay 10 USD for 11 apples. Don't forget your X. Using $X$ we redefined, this can be written as $11X = 10$, so dividing by $11$ both sides, we get $X = 10/11$. The conclusion from this paragraph is: the price of one apple if we buy 11 apples from from Store B is $10/11$ USD.
The problem comes down to comparing the following two fractions:
Review the last lesson about comparing fractions. We need to make the same positive bottom and then compare the tops. Note that
$$\frac{9}{10} = \frac{9 \cdot 11}{10 \cdot 11} = \frac{99}{10 \cdot 11},$$
and
$$\frac{10}{11} = \frac{10 \cdot 11}{10 \cdot 11} = \frac{110}{10 \cdot 11}.$$
Hence, comparing the tops of the right-hand sides (which have the same positive bottom $10 \cdot 11$), we realize that Store A sells the apple at a cheaper price. This is the answer.
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 USD), so
$$(0.9)X = 10.$$
Don't forget your $X$! We are trying to solve for $X$. How do we manipulate this to get $X$? The easiest way is to multiply $10$ on both sides. Then we get $9X = 100$. Then we divide both sides by $9$ to get $X = 100/9$ (USD). If you like decimal expressions, the answer would be $X = 11.111111111...$ (USD), but there is no way we can stop this expression, so we either have to keep $...$ or say $100/9$ USD is the answer.
Remark. Let me provide another example, but it turns out to be slightly harder than I wished. However, this example is certainly within the range of GMAT's difficulty, so take a deep breath and study it when you have enough energy!
Example (Hard). Store A and B sell apples. Originally, the price of one apple from either store was 1 USD. However, the stores started to compete each other by starting their discount programs. Store A gives you 10% discount when you buy 10 apples. On the other hand, Store B gives you one more apple when you buy 10 apples. If we compare the price per one apple which store sells apples cheaper if we purchase 10 apples (meaning 11 total apples for Store B)?
Remark. I really like this problem, because when I first tried it, I thought both stores sold the apples at the same price. As you will see, it turns out they didn't, and I think it captures a great strategy among marketers: a customer can feel that they are purchasing at a cheaper price even if they don't. This strategy is used in many places, and one instance where I experience this is Starbucks Rewards program.
How do we start? Don't forget your X! You first need to set $X$ to be something you want to find. First, we want to find the price of one apple Store A sells when we buy 10 apples from them, so we let
$$X = \text{ price of one apple when we buy 10 apples from Store A}.$$
How do we compute this? This is where much thinking comes in. It is difficult to find $X$ right away, but it is easier to find $10X$ because that is the price of 10 apples from Store A. Originally this was supposed to be 10 USD, but under their discount program, we get 10% discount. We need to recall that 10% means $1/10$ because 100% means $1$. Since Store A cuts out $1/10$ of the price, what we need to pay for 10 apples is $9/10$ of 10 USD. Thus, we have
$$10X = \frac{9}{10} \cdot 10,$$
where the unit is given by USD, but we chose to suppress it. The right-hand side can be manipulated as
$$\frac{9}{10} \cdot 10 = \frac{9}{10} \cdot \frac{10}{1} = \frac{9 \cdot 10}{1 \cdot 10} = \frac{9}{1} = 9.$$
Combining this with the previous equation, we have $10X = 9$, so dividing this by $10$ both sides, we have $X = 9/10$. Don't forget what your $X$ meant: it meant the price of one apple if you buy 10 apples from Store A. The conclusion so far is: the price of one apple if we buy 10 apples from from Store A is $9/10$ USD.
Now, we need to do the same procedure for Store B. Your $X$ needs to be what you want to find, so we set
$$X = \text{ price of one apple when we buy 11 apples from Store B}.$$
First, carefully note that now we work with 11 apples rather than 10 apples because this is what problem said in the parenthetical remark. How do we find this X? You need to think quite a bit and read the problem over and over until you have some sense of where to start. One way to start is to noting that it is easier to find the price of 11 (total) apples from Store B. Originally it is supposed to be 11 USD, but one of them comes free, so we pay 10 USD for 11 apples. Don't forget your X. Using $X$ we redefined, this can be written as $11X = 10$, so dividing by $11$ both sides, we get $X = 10/11$. The conclusion from this paragraph is: the price of one apple if we buy 11 apples from from Store B is $10/11$ USD.
The problem comes down to comparing the following two fractions:
- $9/10$ for Store A and
- $10/11$ for Store B.
Review the last lesson about comparing fractions. We need to make the same positive bottom and then compare the tops. Note that
$$\frac{9}{10} = \frac{9 \cdot 11}{10 \cdot 11} = \frac{99}{10 \cdot 11},$$
and
$$\frac{10}{11} = \frac{10 \cdot 11}{10 \cdot 11} = \frac{110}{10 \cdot 11}.$$
Hence, comparing the tops of the right-hand sides (which have the same positive bottom $10 \cdot 11$), we realize that Store A sells the apple at a cheaper price. This is the answer.
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