Complete Guide to Fractions and Ratios in ACT Math

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).

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.

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.

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?

1. $3{2/15}$
2. $3{2/3}$
3. $4$
4. $4{4/15}$
5. $4{1/3}$

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, $3{2/3}$

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.

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.)

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 $3:2$

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.

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?

1. 12
2. 28
3. 84
4. 112
5. 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).

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.

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.

1)

2)

3)

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

1. Fewer than 3
2. 3
3. 6
4. 7
5. More than 7

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).