Ratios & Percents - ACT Math

To convert between minutes and hours, you can use dimensional analysis to ensure that you are performing the right calculations. You know that there are 60 minutes for every 1 hour, and you start with minutes and need to get your answer in terms of hours. That means that you want to set up the equation so that the units "minutes" cancels and leaves you with just "hours":

You can see here that "minutes" is in both numerator and denominator, allowing you to cancel those units. And it dictates that to get your answer you'll divide 450 by 60. That reduces as a fraction to 45 divided by 6, which comes out to 7.5.

When converting between units, it can be helpful to use dimensional analysis to set up your equation. You know that there are 12 inches in 1 foot, and that you need to convert 18 feet to a number of inches. If you then structure your math so that the units cancel via division, you can ensure that you're making the right "do I multiply or do I divide by 12" decision:

Helps you determine that you need to multiply, because then the "feet" units cancel leaving you with the units you want, which is inches. This means you multiply 18 by 12, which gives you 216 inches.

A great way to ensure you make the proper "do I multiply or divide?" decision on any conversion problem is to use dimensional analysis, which means to set up the units in your calculation so that the units you don't want in your answer cancel, and the unit(s) you do want in your answer remain. Here you're given gallons and asked to convert to quarts, so you want to set up the math so that gallons cancels. That means you'll use:

This preserves the ratio of 4 quarts to every 1 gallon and allows the unit "gallons" to cancel via division so that you're left with quarts. That also means that you multiply 8 by 4 to get the answer, 32.

When you're working with conversion problems, it is helpful to use dimensional analysis to make sure that you're making the proper "do I multiply or divide by this conversion" decision. Here you're given a number of miles (she wants to ride 20 miles) and you need to get to kilometers. So you can structure your math with miles in the denominator of the fraction, and that means that the units "miles" will cancel leaving you with just "kilometers":

As you can see, this tells you to divide by 0.62, and when you've canceled the unit "miles" you're left with just "kilometers." Note, also, that the question asks for an estimate ("which of the following is closest") so you can safely divide by just 0.6, or 3/5, to get to 33.33, and the only answer that is close is 32.

When you're working with conversion problems, it is helpful to use dimensional analysis to ensure that you are properly applying the conversion (i.e. "should I multiply or divide?"). Here you're given a number of steps and you need to convert to miles using the conversion 2,500 steps = 1 mile. You can then set up your math so that the unit "steps" cancels and you're left with just "miles." That would mean:

As you can see, steps will cancel due to division, meaning that you'll divide 320,000 by 2,500 and be left with the proper unit miles. To do this math by hand, you should also see that you can factor out 100 from both 320,000 and 2,500, leaving you with the problem 3200 divided by 25. This leads you to the correct answer, 128.

Whenever you're working with unit conversions, it is a good idea to use dimensional analysis to structure your math to help you choose the right operation (multiply vs. divide). This means that you'll set up your equation to cancel the units that you don't want in your answer, and therefore you'll be left with the proper units. Here you're given inches and asked to convert to centimeters, so you'll set up the math with inches in the denominator so that the units cancel:

This means that you'll multiply 10 by 2.54, and the inch units will cancel ensuring that you're properly converting to centimeters. 10 times 2.54 is 25.4, which rounds to 25.

Whenever you're working with unit conversions, it is a good idea to use dimensional analysis to ensure that you are setting up your calculation - specifically the choice of whether you multiply or divide by the conversion ratio - properly. This means that you'll set up the math so that the units you're given cancel, leaving you with the units you need to arrive at in the end. Here you're given kilograms and want to get to pounds, so you'll multiply by a conversion with kilograms in the denominator so that kilograms cancel, leaving you with pounds:

This means that you multiply 25 by 2.2 and that kilograms cancel and the only unit left is the one you want, pounds. 25 times 2.2 is 55, giving you the correct answer.

Whenever you're working with unit conversions, it is a good idea to use dimensional analysis to structure your math properly - specifically as it pertains to the question of whether you should multiply or divide by the conversion ratio provided. That means you'll set up the math so that the units you don't want cancel, leaving you with the units you do. Here you're given pints and asked to arrive at quarts, so you'll set up your math such that pints is in the denominator of the conversion ratio so that it cancels. That means that your math will look like:

This means you'll divide 16 by 2 to get your answer, and since pints cancels via division, you know that you've converted properly to quarts. 16 divided by 2 is 8, making 8 the correct answer.

Whenever you're working with unit conversions, it is a good idea to use dimensional analysis to ensure that you are structuring your calculation properly, as often the decision of whether to multiply or divide by the conversion ratio can be tricky. That means that you'll set up the math so that the units you don't want in your answer (here that's feet) will cancel, leaving you with the units you do want (here that's inches). Your math here would look like:

Feet here will cancel via division, leaving you with inches. And this tells you to multiply 18 by 12 giving you an answer of 216.

Whenever you're facing a unit conversion problem it is a good idea to use dimensional analysis to help you structure the math - whether you should multiply or divide by the provided conversion ratio - properly. That means that you'll set up the math such that the units you don't want in your answer cancel, leaving you with the units you do want. Here you're given quarts but asked to convert to pints, so you'll set up the math so that quarts are in the denominator and cancel, leaving you with pints:

This means that quarts cancel leaving you with pints, and tells you that you have to multiply 12 by 2. The correct answer is 24.

In any percent problem, it is crucial that you know exactly which value the percent is to be taken of. Here Portia is overcharged by 20%, meaning that she paid 20% more than the item should have cost. That means that the amount she paid, $480, equals 20% more than she should have paid:

, where is the price she should have paid.

To solve for , divide both sides by :

This means that she should have paid $400 but instead paid $480, so she overpaid by $80.

Note that a common mistake on percent problems is taking the percent of the wrong value: here you might be tempted to simply take 20% of 480, for example. The ACT test makers know that, and will often set up problems so that you're given the result of the percent change (here the result is Portia paying $480) and need to work your way back to the start value as you did here. Always pause to ensure that you know which value (or variable) gets multiplied by the percent change.

This problem rewards students who do the work rather than just looking at a problem and going with what “looks right.” Whenever you see a problem that asks you to find the percent difference between two unknowns, you can probably pick your own numbers to make the problem easier since the problem is asking for a relationship rather than a value.

In this case, since you are dealing with percents, it is a good idea to start with the number 100. If you increase 100 by 10%, you will get

after the first increase.

During the second week, you are told, the price of the stock increases by 10% of the new value. Be careful here to take the percent of the correct value, 110, rather than the original value. If you increase 110 by 10%, you get:

.

To find the percent increase from there, you find the difference and then divide by the original before multiplying by 100:

Remember that when you are dealing with taking the percent of a number and then taking the percent of the result, that you must be careful to take the percent of the correct number. In this case, that means increase $60 by 50% and then taking 20% off that new number. What you cannot do is either take off 20% of 60 or just increase 60 by 30% rather than doing each step in order. Doing either will lead to a trap that the Testmaker has left for you.

The first step you’re given is to increase $60 by 50%. This is the same as taking 150% of $60, or as multiplying it by 1.5. If you do this you get;

From there, you now need to find what 20% off 90 is in order to get your solution. You can do this by first finding 10% of 90 by moving the decimal point one place to the left to get 9 and then doubling it to get 18.

Subtracting 18 from 90 gives that a customer would pay $72.

With percent change problems it is important to recognize that you take the percentage of the original value. The formula is , and then multiply by 100 to convert to a percent.

Recognizing that, the percent change here is . This is the type of calculation you should be able to do quickly in your head: to go from 80 to 100, a number must increase by one-quarter of itself, which is 25%.

This percents problem can be set up by noting the height of the tree () in two ways:

(the tree is 80% taller than Dan)

(the tree is 20% taller than Alex)

Now you can set the equations equal, since each defines the value of :

With this, the denominators cancel (you can just multiply both sides of the equation by 5), leaving:

Divide both sides by and you'll have your answer:

, which simplifies to: .

This means that Alex is 50% taller than Dan.

A critical element of all percent problems is that you must pay particular attention to what the percent is being taken OF. Here, recognize that you're given two percents in terms of the beginning (start of 2006) value, but you're asked for the percent change for the year 2007, meaning that you're looking for the percent increase in terms OF the start of 2007 value.

If you say that the initial value at the beginning of 2006 was , then you know that at the end of 2006 / beginning of 2007 the value was , and at the end of 2007 it was . What you're being asked, then is the percent change from when it was at to when it was . The difference is , so you can use the calculation . This means that you're looking at a fraction of , which translates to 12.5%.

Alternatively, you could pick numbers to make this easier. If you say that the land was worth $100 to start, grew to $160 by the end of the first year, and then was worth $180 at the end of the second year, then you know that it gained $20 from a base value of $160 over the course of the second year, again for a 12.5% increase.

Algebraically, if Dinesh started at miles, he increases to miles in the second week, and decreases to for the final week -- a 10 percent decrease from his starting point.

You can set this problem up algebraically by recognizing what you're really solving for, which is essentially a ratio of doughnuts () to total (). You're given that the ratio of glazed () to doughnuts () is 70/100, and you're given that glazed () to total () is 21/100. Knowing that, you can set up:

and

And since has to be equal in each of those equations, you can set the equations equal:

From there, your goal is to solve for , so you'll divide both sides by to get:

And then multiply both sides by to cancel the fraction on the left and move all the numbers to the right. Now you have:

Now you should see how cleanly the terms on the right factor, leaving you with:

. If out of is out of , that means that the answer is .

With percent problems, the key is often to make sure that you take the percent of the correct value. In this case, the initial 25% off means that customers will pay 75% of the original price. Then for the second discount, keep in mind that the discount is taken off of the sale price, not of the original price. So that's 20% off of the 75% that they did pay, which can be made easier by looking at what the customer does pay: 80% of the 75% sale price. Using fractions, that means they pay: of the original price, which nets to of the original price, or 60%.

We can set up equations for income before and after the wage reduction. Initially, the employee earns wage and works hours per week. After the reduction, the employee earns wage and works hours. By setting these equations equal to each other, we can determine the increase in hours worked: (divide both sides by ) We know that the new number of hours worked will be 25% greater than the original number.