Fractions and ratios (and by extension rational numbers) are all around us and, knowingly or not, we use them every day. If you wanted to brag over the fact that you ate half a pizza by yourself (and why not?) or you needed to know how many parts water to rice you need when making rice on the stove (two parts water to one part rice), then you need to communicate this using fractions and ratios.

In essence, fractions and ratios represent pieces of a whole by comparing those pieces either to each other or to the whole itself. Don’t worry if that sentence makes no sense right now. We’ll break down all the rules and workings of these concepts throughout this guide--both how these mathematical concepts work in general and how they will be presented to you on the ACT.

**Whether you are an old hat at dealing with fractions, ratios, and rationals, or a novice, this guide is for you. **This guide will break down what these terms mean, how to manipulate these kinds of problems, and how to answer the most difficult fraction, ratio, and rational number questions on the ACT.

## What are Fractions?

$${\a\piece}/{\the\whole}$$

Fractions are pieces of a whole. They are expressed as the amount you have (the numerator) over the whole (the denominator).

Amy’s cat gave birth to 8 kittens. 5 of the kittens had stripes and 3 had spots. What fraction of the litter had stripes?

$5/8$ of the litter had stripes. 5 is the numerator (top number) because that was the amount of striped kittens, and 8 is the denominator (bottom number) because there are 8 kittens total in the litter (the whole).

*Kitten math is the best kind of math.*

### Special Fractions

There are several different kinds of "special fractions" that you must know in order to solve the more complex fraction problems. Let us go through each of these:

**A number over itself equals 1**

$6/6 = 1$

$47/47 = 1$

${xy}/{xy} = 1$

**A whole number can be expressed as itself over 1**

$17 = 17/1$

$108 = 108/1$

$xy = {xy}/1$

**0 divided by any number is 0**

$0/0 = 0$

$0/5 = 0$

$0/{xy} = 0$

** **

**Any number divided by 0 is undefined**

Zero cannot act as a denominator. For more information on this check out our guide to advanced integers. But, for now, all that matters is that you know that 0 cannot act as a denominator.

** **

*Now let's find out how to manipulate fractions until we unlock the answers we want.*

### Reducing Fractions

If you have a fraction in which both the numerator and the denominator can be divided by the same number (called a “common factor”), then the fraction can be reduced. Most of the time, your final answer will be presented in its most reduced form.

In order to reduce a fraction, you must find the common factor between each piece of the fraction and divide both the numerator and the denominator by that same amount. By dividing both the numerator and the denominator by the same number, you are able to maintain the proper relationship between each piece of your fraction.

So if your fraction is $5/25$, then it can be written as $1/5$. Why? Because both 5 and 25 are divisible by 5.

$5/5 = 1$

And

$25/5 = 5$.

So your final fraction is $1/5$.

** **

### Adding or Subtracting Fractions

You can add or subtract fractions as long as their denominators are the same. To do so, you keep the denominator consistent and simply add the numerators.

$2/11 + 6/11 = 8/11$

But you CANNOT add or subtract fractions if your denominators are unequal.

$2/11 + 4/5 = ?$

So what can you do when your denominators are unequal? You must *make* them equal by finding a common multiple (number they can both multiply evenly into) of their denominators.

$2/11 + 4/5$

Here, a common multiple (a number they can both be multiplied evenly into) of the two denominators 11 & 5 is 55.

To convert the fraction, you must multiply both the numerator and the denominator by the amount the denominator took to achieve the *new* denominator (the common multiple).

Why multiply both? Just like when we reduced fractions and had to divide the numerator and denominator by the same amount, now we must *multiply* the numerator and denominator by the same amount. This process keeps the fraction (the relationship between numerator and denominator) consistent.

To get to the common denominator of 55, $2/11$ must be multiplied by $5/5$. Why? Because $11 * 5 = 55$.

$(2/11)(5/5) = 10/55$.

To get to the common denominator of 55, $4/5$ must be multiplied by $11/11$. Why? Because $5 *11 = 55$.

$(4/5)(11/11) = 44/55$.

Now we can add them, as they have the same denominator.

$10/55 + 44/55 = 54/55$

We cannot reduce $54/55$ any further as the two numbers do not share a common factor.

So our final answer is $54/55$.

Here, we are not being asked to actually add the fractions, just to find the least common denominator so that we *could* add the fractions.

Because we are being asked to find the least amount of something, we should start at the smallest number and work our way down (for more on using answer choices to help solve your problem in the quickest and easiest way, check out our article on plugging in answers).

Answer choice A is eliminated, as 40 is not evenly divisible by 12.

120 is evenly divisible by 8, 12, and 15, so it is our least common denominator.

So **our final answer is B**, 120.

** **

### Multiplying Fractions

Luckily it is much simpler to multiply fractions than it is to add or divide them. There is no need to find a common denominator when multiplying--you can just multiply the fractions straight across.

To multiply a fraction, first multiply the numerators. This product becomes your new numerator.

Next, multiply your two denominators. This product becomes your new denominator.

$2/3 * 3/4 = (2 * 3)/(3 * 4) = 6/12$

And again, we reduce our fraction. Both the numerator and the denominator are divisible by 6, so our final answer becomes:

$1/2$

**Special note:** you can speed up the multiplication and reduction process by finding a common factor of your cross multiples before you multiply.

$2/3 * 3/4$ => $1/1 * 1/2$ => $1/2$.

Both 3’s are multiples of 3, so we can replace them with 1 ($3/3 = 1$).

Our other cross multiples are 2 and 4, which are both multiples of 2, so we were able to replace them with 1 and 2, respectively ($2/2 = 1$ and $4/2 = 2$).

Because our cross multiples had factors in common, we were able to reduce the cross multiples before we even began. This saved us time in reducing the final fraction at the end.

**Take note that we can only reduce cross multiples when ***multiplying*** fractions, never while adding or subtracting them!** It is also a *completely optional step*, so do not feel obligated to reduce your cross multiples--you can always simply reduce your fraction at the end.

** **

### Dividing Fractions

In order to divide fractions, we must first take the reciprocal (the reversal) of one of the fractions. Afterwards, we simply multiply the two fractions together as normal.

Why do we do this? Because division is the opposite of multiplication, so we must reverse one of the fractions to turn it back into a multiplication question.

${1/3} ÷ {3/8} => {1/3} * {8/3}$ (we took the reciprocal of $3/8$, which means we flipped the fraction upside down to become $8/3$)

${1/3} * {8/3} = 8/9$

*Now that we've seen how to solve a fraction problem the long way, let's talk short cuts.*

### Decimal Points

Because fractions are pieces of a whole, you can also express fractions as either a decimal point or a percentage.

To convert a fraction into a decimal, simply divide the numerator by the denominator. (The $/$ symbol also acts as a division sign)

$3/10 => 3 + 10 = 0.3$

Sometimes it is easier to convert a fraction to a decimal in order to work through a problem. This can save you time and effort trying to figure out how to divide or multiply fractions.

This is a perfect example of a time when it might be easier to work with decimals than with fractions. We’ll go through this problem both ways.

**Fastest way--with decimals:**

Simply find the decimal form for each fraction and then compare their sizes. To find the decimals, divide the numerator by the denominator.

$5/3 = 1.667$

$7/4 = 1.75$

$6/5 = 1.2$

$9/8 = 1.125$

We can clearly see which fractions are smaller and larger now that they are in decimal form. In ascending order, they would be:

$1.125, 1.2, 1.667, 1.75$

Which, when converted back to their fraction form, is:

$9/8, 6/5, 5/3, 7/4$

So **our final answer is A**.

**Slower way--with fractions:**

Alternatively, we could compare the fractions by finding a common denominator of each fraction and then comparing the sizes of their numerators.

Our denominators are: 3, 4, 5, & 8.

We know that there are no multiples of 4 or 8 that end in an odd number (because an even number * an even number = an even number), so a common denominator for all must end in 0. (Why? Because all multiples of 5 end in 0 or 5.)

Multiples of 8 that end in 0 are also multiples of 40 (because $8 * 5 = 40$). 40 is not divisible by 3 and neither is 80, but 120 is.

120 is divisible by all four digits, so it is a common denominator.

Now we must find out how many times each denominator must be multiplied to equal 120. That number will then be the amount to which we multiply the numerator in order to keep the fraction consistent.

$120/3 = 40$

$5/3$ => ${5(40)}/{3(40)}$ => $200/120$

$120/4 = 30$

$7/4$ => ${7(30)}/{4(30)}$ => $210/120$

$120/5 = 24$

$6/5$ => ${6(24)}/{5(24)}$ => $144/120$

$120/8 = 15$

$9/8$ => ${9(15)}/{8(15)}$ => $135/120$

Now that they all share a common denominator, we can simply look to the size of their numerators and compare the smallest and the largest. So the order of the fractions from least to greatest would be:

$135/120, 144/120, 200/120, 210/120$

Which, when converted back into their original fractions, is:

$9/8, 6/5, 5/3, 7/4$

So once again, **our final answer is A**.

As you can see, we were able to solve the problem using either fractions *or* decimals. How you chose to approach these types of problems is completely up to you and depends on how you work best, as well as your time management strategies.

### Percentages

After you convert your fraction to a decimal, you can also turn it into a percentage (if the need arises).

To get a percentage, multiply your decimal point by 100.

So 0.3 can also be written as 30%, because $0.3 * 100 = 30$.

0.01 can be written as 1% because $0.01 * 100 = 1$, etc.

*Be mindful of your decimals and percentages and don't mix them up! 0.1 is NOT the same thing as 0.1%. *

### Mixed Fractions

Sometimes you may be given a mixed fraction on the ACT. A mixed fraction is a combination of a whole number and a fraction.

For example, $5{1/3}$ is a mixed fraction. We have a whole number, 5, and a fraction, $1/3$.

You can turn a mixed fraction into an ordinary fraction by multiplying the whole number by the denominator and then adding that product to the numerator. The final answer will be ${\the \new \numerator}/{\the \original \denominator}$.

$5{1/3}$

$(5)(3) = 15$

$15 + 1 = 16$

So your final answer = $16/3$

You must convert mixed fractions into non-mixed fractions in order to multiply, divide, add, or subtract them with other fractions.

A cobbler charges a flat fee of 45 dollars plus 75 dollars per hour to make a pair of shoes. How many hours of labor was spent making the shoes if the total bill was $320?

- $3{2/15}$
- $3{2/3}$
- $4$
- $4{4/15}$
- $4{1/3}$

If the total bill was 320 dollars and the flat fee was 45 dollars, we must subtract the flat fee from the total bill in order to find the number of hours the cobbler worked.

$320 - 45 = 275$

So the cobbler worked 275 dollars’ worth of hours. In order to find out how many hours that is, we must divide the earnings by the hourly fee.

$275/75 = 3{50/75}$

75 was able to go evenly into 225, leaving 50 out of 75 left over.

Because 50 and 75 share a common denominator of 25, we can reduce $3{50/75}$ to:

$3{2/3}$

So **our final answer is B**, $3{2/3}$

*Now that we've broken down all there is to know about ACT fractions, let's take a look at their close cousin--the ratio.*

## What are Ratios?

Ratios are used as a way to compare one thing to another (or multiple things to one another).

If Piotr has exactly 2 grey scarves and 7 red scarves in a drawer, the ratio of grey scarves to red scarves is 2 to 7.

### Expressing Ratios

Ratios can be written in three different ways:

$A \to B$

$A:B$

$A/B$

No matter which way you write them, these are all ratios comparing A to B.

*Most all chemical molecules are named for their ratios. Here, one of our products is carbon dioxide (one part carbon, two parts oxygen).*

### Different Types of Ratios

Just as a fraction represents a part of something out of a whole (written as: ${\a \part}/{\the \whole}$), a ratio can be expressed as either:

${\a \part}:{\a \different \part}$

OR

$\a \part:\the \whole$

Ratios compare values, so they can either compare individual pieces to one another or an individual piece to the whole.

If Piotr has exactly 2 grey scarves and 7 red scarves in a drawer, the ratio of grey scarves to *all* the scarves in the drawer is 2 to 9. (Why 9? Because there are 2 grey and 7 red scarves, so together they make $2 + 7 = 9$ scarves total.)

### Reducing Ratios

Just as fractions can be reduced, so too can ratios.

Danielle collects toy racecars. 12 of them are blue and 4 of them are yellow. What is the ratio of of blue cars to yellow cars in her collection?

Right now, the ratio is $12:4$. But they have a common denominator of 4, so this ratio can be reduced.

$12/4 = 3$

$4/4 = 1$

So the cars have a ratio of $3:2$

### Increasing Ratios

Because you can reduce ratios, you can also do the opposite and *increase* them. In order to do so, you must multiply each piece of the ratio by the same amount (just as you had to divide by the same amount on each side to reduce the ratio).

So the ratio of $3:2$ can also be

$3(2):2(2) = 6:4$

$3(3):2(3) = 9:6$

$3(4):2(4) = 12:8$

And so on.

Though this presents itself as a geometry problem, we don’t need to know any geometry in order to solve it--we only need to know about ratios.

We have two triangles in a ratio of 2:5 and the smaller triangle has a hypotenuse of 5 inches. This means that we need to increase each side of the ratio by the amount it takes 2 to go into 5.

$5/2 = 2.5$

So we must increase each side of the ratio by a matter of 2.5

$2(2.5):5(2.5)$

$5:12.5$

Our new, increased ratio is 5:12.5, which means that the larger hypotenuse is 12.5.

**Our final answer is K.**

*Expand ratios, reduce them--go wild!*

### Finding the Whole

If you are given a ratio comparing two parts ($\piece:\another \piece$), and you are told to find the whole amount, simply add all the pieces together.

It may help you to think of this like an algebra problem wherein each side of the ratio is a certain multiple of x. Because each side of the ratio must always be divided or multiplied by the same amount to keep the ratio consistent, we can think of each side as having the same variable attached to it.

For example, a ratio of $6:7$ can be:

$6(1):7(1) = 6:7$

$6(2):7(2) = 12:14$

And so on, just as we did above.

But this means we could also represent $6:7$ as:

$6x:7x$

Why? Because each side must change at the same rate. And in this case, our rate is $x$.

So if you were asked to find the total amount, you would add the pieces together.

$6x + 7x = 13x$. The total amount is $13x$. In this case, we don’t have any more information, but we know that the total MUST be either 13 or any number divisible by 13.

So let’s take a look at another problem.

Clarissa has a jewelry box with necklaces and bracelets. The necklaces and bracelets are in a ratio of 4:3. What is NOT a possible number of total pieces of jewelry Clarissa can have in the box?

- 12
- 28
- 84
- 112
- 140

In order to find out how many pieces of jewelry she may have total, we must add the two pieces of our ratio together.

So $4x + 3x = 7x$

This means that the total number of jewelry items in the box has to either be 7 or any multiple of 7. Why? Because $4:3$ is the most reduced form of the ratio of jewelry items in the box. This means she could have:

$4(1):3(1) = 7$ jewels in the box (7 jewelry pieces total)

$4(2):3(2) = 8:6$ jewels in the box (14 jewelry pieces total)

$4(3):3(3) = 12:9$ jewels in the box (21 jewelry pieces total)

And so forth. We don’t know exactly how many jewelry items she has, but we know that it must be a multiple of 7.

**This means our answer is A**, 12. There is no possible way that she can have 12 jewels in the box, because 12 is not a multiple of 7 and one cannot have half a bracelet (unless something has gone terribly wrong).

You may also be asked to find the number of individual pieces in your ratio after you are given the whole. This is exactly the opposite of what we did above.

The is the exact same process as finding the whole, but in reverse.

We know we must add the pieces of our ratio to find our multiple of 30. And we also know our ratio is $2:3$. So let us add these together.

$2x + 3x = 5x$

Together, our ratio components add up to $5x$. And there are 30 feet total. So:

$30/5 = 6$

$x = 6$

This means that we must multiply each side of our ratio by 6 in order to get the exact amount of wood used. This means that each piece is:

$2(6):3(6)$

$12:18$

Which means our shorter piece is 12 feet long.

**Our final answer is H**, 12.

*And now we come to rational and irrational numbers.*

## Rational and Irrational Numbers

A rational number is any number that can be written as a fraction of two integers (where the denominator does NOT equal to 0).

All other numbers are considered irrational.

**Rational Numbers:**

$7/2, 5, 1/212, 0.66666667$

Why is 5 a rational number? Because it can be expressed as the fraction $5/1$.

Why is 0.6666667 a rational number? Because it can be expressed as the fraction $2/3$

** **

**Irrational Numbers:**

$π, √2, √3$

Why is $π$ irrational? Because there is no fraction of two integers that can properly express it (through 22/7 comes awfully close).

(Hint: if the decimals continue on forever *without* repeating, the number is irrational)

Here, we are being asked to find the single rational number. Even if you didn’t know what a rational number meant, you might be able to figure this problem out just by finding the answer choice that stands out the most. But since you DO know what rational and irrational numbers are, it makes the problem even easier.

Many square roots are irrational (unless they are roots of perfect squares like $√16 = 4$). We can immediately eliminate answer choices A, B, and C, as they are not perfect squares and so are irrational.

We can also eliminate answer choice D. When we reduce the fraction, we get $√{1/5}$, and this would also get us an irrational number.

This leaves us with answer choice E.

We can see that both the numerator and the denominator of the fraction $64/49$ inside the square root sign are perfect squares. Since the fraction is under the root sign, let us take the square root of each of these.

So our final fraction would look like:

$√{64/49}$ => $8/7$

Because our final fraction is represented as a fraction with two integers, this is a rational number.

So **our final answer is E**.

*So let's break down how to solve these kinds of questions when they show up on the test.*

## How to Solve Fraction, Ratio, and Rational Number Questions

When you are presented with a fraction or ratio problem, take note of these steps to find your solution:

**1) Identify whether the problem involves fractions or ratios**

A fraction will involve the comparison of a $\piece/\whole$.

A ratio will almost always involve the comparison of a $\piece:\piece$ (or, *very rarely*, a $\piece:\whole$).

You can tell when the problem is ratio specific as the question text will do one of three things:

- Use the : symbol,
- Use the phrase “___ to ___”
- Explicitly use the word “ratio” in the text.

If the questions wants you to give an answer as a ratio comparing two pieces, make sure you don’t confuse it with a fraction comparing a piece to the whole!

** **

**2) If a ratio question asks you to change or identify values, first find the sum of your pieces**

In order to determine your total amount (or the non-reduced amount of your individual pieces), you must add all the parts of your ratio together. This sum will either be your complete whole or will be a factor of your whole, if your ratio has been reduced.

**3) When in doubt try to use decimals**

Decimals can make it much easier to work out problems rather than using fractions. So do not be afraid to convert your fractions into decimals to get through a problem more quickly and easily.

**4) Remember your special fractions**

Always remember that a number over 1 is the same thing as the original number, and that when you have a number over itself, it equals 1.

*Get ready, get set...GO!*

## Test Your Knowledge

**1)**

**2)**

**3)**

**4)** How many irrational numbers are there between 1 and 8?

- Fewer than 3
- 3
- 6
- 7
- More than 7

**Answers:** B, J, D, E

** **

**Answer Explanations:**

**1)** For this problem, we must combine our like terms in order to eventually isolate $k$ (for more on this, check out our guide to ACT single variable equations).

We know that, when adding fractions, we must give them the same denominator, so we can manipulate our fractions to have matching denominators and solve from there.

Alternatively, we could again use decimal points instead of fractions. We will go through both ways here.

**Method 1--Fractions**

We have ${1/3}k$ and ${1/4}k$ that we must add. They share a common multiple of 12, so let us convert them to fractions out of 12.

$1/3$ => ${1(4)}/{3(4)}$ => $4/12$

$1/4$ => ${1(3)}/{4(3)}$ => $3/12$

Now that they have the same numerator, we can combine them to be:

$4/12 + 3/12 = 7/12$

So our equation is:

${7/12}k = 1$

Now we must divide both sides by $7/12$, which means that we must inverse and multiply.

$k = 1(12/7)$

$k = 12/7$

**So our final answer is B. **

**Method 2--Decimals**

Instead of using and converting fractions, we also could have used decimals instead.

$1/3$ => $0.333$

$1/4$ => $0.25$

Because they are decimals, we can simply add them together to be:

$0.333k + 0.25k = 0.583k$

$0.58k = 1$

$k = 1/0.583$

$k = 1.715$

Now, simply convert the answer choices to decimals to find one that matches. In this case answer choice A would be far too small, and answers D and E are whole numbers, so they can all be eliminated. Answer choice C would be $7/2 = 3.5$.

This leaves us with answer choice B:

$12/7 = 1.714$

**So our final answer is, again, B. **

**2)** This question specifically asks for a rational number answer, but it is a bit deceptive, as a quick glance shows us that *all* the answer choices are rational numbers. This means you can ignore this stipulation for the time being.

Again, we can solve this problem in one of two ways--via fractions or via decimals. We will go through both methods.

**Method 1--Fractions**

We are trying to find a rational fraction halfway between $1/5$ and $1/3$, so let us convert them into fractions with the same denominator.

A common multiple of 3 and 5 is 15, so let us make that their new denominator.

$1/5$ => ${1(3)}/{5(3)}$ => $3/15$

$1/3$ => ${1(5)}/{3(5)}$ => $5/15$

Well the rational number exactly halfway between $3/15$ and $5/15$ is $4/15$.

So **our answer is J,** $4/15$.

**Method 2--Decimals**

Again, if fractions aren't your favorite, you can always feel free to use decimals.

First, convert $1/5$ and $1/3$ into decimals.

$1/5 = 0.2$

$1/3 = 0.333$

Now, find the decimal halfway between them:

${0.2 + 0.333}/2 = 0.2665$ (For more on this process, check out our guide to ACT mean, median, and mode)

Now, let us find the answer choice that, when converted into a decimal, matches our answer.

If you know your decimals, then you know that $1/2 = 0.5$ and $1/4 = 0.25$, so these can be eliminated.

We are now left with $2/15$, $4/15$, and $8/15$. The smart thing to do here is to pick the middle value and then go up or down if the mid value is too small or too large. So if we test $4/15$, we get:

$4/15 = 0.2666$

Success! We nailed it at the mid value, no need to try the others.

**Our final answer is, again, J.**

**3)** Even though this problem may, at first glance, look like a fraction problem, it is a ratio problem. We can tell because the question specifically asks for the *ratios* of the boys' sandwich consumption.

If you're not paying attention, you can easily make a mistake and treat the question as a fraction problem when ratios are written using the "/" symbol.

So we have Jerome, who eats half the sandwich and Kevin, who eats one third, and Seth, who eats the rest. Now we can do this problem several ways, but let us pick two of the most straightforward--ratio and fraction manipulation or plugging in your own numbers (for more on this strategy, check out guide to plugging in numbers).

**Method 1--Ratio and Fraction Manipulation**

Because we are not told the portion of the sandwich that Seth ate, we must find it. Fractions represent pieces of the whole and the whole is 1 (because anything over itself = 1). So let us add our two fractions and subtract that sum from 1 to find Seth's share of the sandwich.

$1/2 + 1/3$. First, we must convert these fractions to ones with a shared denominator. Both 2 and 3 are multiples of 6, so we will use 6 as our new denominator.

$1/2$ => ${1(3)}/{2(3)}$ => $3/6$

$1/3$ => ${1(2)}/{3(2)}$ => $2/6$

Now, let us add them together and subtract their sum from 1.

$3/6 + 2/6 = 5/6$

$1 - 5/6 = 1/6$

So Seth ate $1/6$ of the sandwich.

And because these fractions now all share a common denominator, we can simply compare their numerators to find their ratio of sandwich shares (remember, ratios compare parts to other parts).

So the sandwich eating fractions are:

$3/6, 2/6,$ and $1/6$

When we just look at the numerators, the ratio is:

$3:2:1$

**Our final answer is D**, $3:2:1$.

**Method 2--Plugging in Numbers**

Instead of working exclusively with fractions and ratios, let's try the problem again using whole numbers. We know that Jerome ate $1/2$ and sandwich and Kevin ate $1/3$, so let's give the sandwich an actual length value that is a shared multiple of those two numbers (note: our sandwich length does not *have* to be a multiple of 2 and 3--it can be anything we want. It simply makes our lives easier to use a common multiple, as that way we can work with integers.)

So let us say that the sandwich is 12 feet long.

If Jerome ate half of it, then he ate:

$12/2 = 6$ feet of sandwich.

If Kevin ate one third of it, then he ate:

$12/3 = 4$ feet of sandwich.

If we add them together, they ate:

$6 + 4 = 10$ feet of sandwich.

Which means that Seth ate:

$12 - 10 = 2$ feet of sandwich.

Now let us compare their shares of 6, 4, and 2.

$6:4:2$

We know that ratios can be reduced if each of the values shares a common factor. In this case, they can all be divided by 2, so let us reduce the ratio.

$6:4:2$ => $3:2:1$

Again,** our final answer is D**, $3:2:1$

**4)** This question asks you to find the amount of irrational numbers between two real numbers, and the simple answer is that there are infinitely many. (Note: there is also an infinite amount of *rational* numbers between any two real numbers as well!).

Why is this true? Think of it this way:

The square root of 1 is rational, because it equals 1, which can be written as $1/1$.

But the square root of 1.01 is irrational. And so is the square root of 1.02, and the square root of 1.03....None of these numbers can be written as ${\an \integer}/{\an \integer}$ (which you can tell because their decimals continue without repeating), and yet they all sit between 1 and 8 on a number line.

So **our final answer is E**, more than 7 (and, in fact, infinite).

*Hurray and huzzah, you did it!*

## The Take-Aways

Don’t let fractions, ratios, and/or rational numbers intimidate you. Once you’ve mastered the basics behind how they behave, you’ll be able to work your way through many of the toughest fraction and ratio problems the ACT can put in your way

The biggest point to look out for, when dealing with fractions and ratios, is *not to mix them up*! Always pay strict attention to times when you are comparing pieces to pieces or pieces to the whole. Though it can be easy to make a mistake during the test, don’t let yourself lose a point due to careless error.

## What’s Next?

For you, fractions are a breeze, ratios were a snap, and rationals?--Forget about it! Luckily for you, there is plenty more to tackle before test day. We have guides aplenty for the many math topics covered on the ACT, including trigonometry, integers, and solid geometry.

**Running out of time during ACT Math practice?** Check out our article on how to finish your math section before it's pencil's down.

**Don't know what score to aim for?** Make sure you have a good grasp of what kind of score would best suit your goals and current skill level, and how to improve it from there.

**Trying to push your score to the top?** Look to our guide on how to get a perfect score, written by a 36 ACT-scorer.

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Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.

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