How to find rate - ACT Math

Remember that distance = rate x time

For the first route, we can set up an equation where the total distance (36 + 44) equals the rate (65 mph) multiplied by the time:

36 +44 = 65t

80 = 65t

t = 80/65 = 1.23 hrs = 1 hr, 14 min

To find the time taken for the second route, we first figure out the distance traveled by using the Pythagorean Theorem.

We know that the "legs" of the right triangle are 44 and 36, where the hypotenuse is the straightline distance (northwest), directly to the destination:

a2+b2=c2

442+362=c2

3232=c2

c=56.85

56.85=40t

56.85/40=t

t=1.42 hrs=1 hr, 25 min

1 hr. 25 min. – 1 hr. 14 min. = 11 min.

To calculate miles per gallon of gas, we take the 478 miles the motorcycle travels and divide it by the amount of gallons in a full tank of gas, 12 gallons. 478/12 = 39.83, or 39.8 when rounded to the tenths place

For the price, we take the 12 gallons and multiply it by the amount that one gallon costs, $4.25. (12)(4.25) = 51

First we determine how many gallons it will take to travel 3,149 miles with this particular car: 3,149/31=101.58 gallons. The cost of gas per gallon= $3.69, therefore 101.58x$3.69= $374.83.

We know that it takes Max an hour to drive 25 miles. We also know that there are 60 minutes in an hour. Using this information we can create the following ratio:

We are trying to calculate the the amount of time it will take to drive 15 miles. Let's create a proportion and use a variable for the unknown time.

Cross-multiply and solve for the time.

We divide 1,457 miles by 47 miles per gallon to find that 31 gallons of gas are needed for the road trip.

We then multiply the gallons of gas by the cost per gallon to find:

31 x 4.13 = 128.03

First, we need to convert the 1 hour into minutes in order to keep consistent with units -- so, the car averages 32 miles per 60 minutes. Then, a ratio can be set up to solve this: 32 mi / 60 min = x mi / 20 min. Cross multiplying and dividing, we get x = 10.667 miles. Rounding to the nearest tenth, this becomes 10.7 miles.

First, find the total amount of gas necessary for the trip. 6363/63 = 101 gallons (easy to see as 63 * 100 = 6300 + 1 * 63 = 6363). Then multiply the number of gallons by the price per gallon of gasoline, 5.05 * 101 = $510.05 and is your answer (again, easy to see when 5.05 * 100 + 1 * 5.05).

Remember that distance/time=rate, so then:

x/15 = 65

x = 65 * 15

x = 975 miles

The general equation is distance = rate x time. In addition, the distance upstream is the same as the distance downstream. So, rup x tup = rdown x tdown. Be sure to convert minutes to hours because the rate is given in mph (miles per hour).

Therefore, (r + 3)(1/2) = (r – 3)(1) and solve for r.

Note, r + 3 is the downstream rate and r – 3 is the upstream rate

In general for work problems: W1 + W2 = 1 where Work = Rate x Time

Note, 1 represents the completed job assignment.

For example, W1 is the rate that the first person would finish the job multiplied by the time it would take two or more people to finish the job completely.

1/3x + 1/2x = 1 where x is the time it would take for both people to complete the job.

Find a common denominator to add the fractions, then solve for x.

x = 1.2 days for one house, but the questions asks about two houses, so the correct answer is 2.4 days.

In order to get the total amount of gas used in Sarah’s trip, first find how much gas was used on the highway and add it to the amount used in the city. Highway gas usage can be found by dividing

and city usage can be found by dividing

.

Then we add these two answers together and get

We know that the rate of a car can be written in the equation:

This means that you need the distance and time of your total trip. We know that the trip was a total of or hours. The distance is easily calculated by multiplying each respective rate by its number of hours, thus, you know:

Therefore, you know that the rate of the total trip was:

Recall that the basic form for a rate is:

, where is generically the amount of work done. Since the question asks for the answer in gallons per hour, you should start by changing your time amount into hours. This is done by multiplying by to get .

Thus, we know:

The rate of draining is once the new pipe is added. Recall that:

, where is the total work output. For our data, this means the total amount of water. Now, we are measuring our rate in hours, so we should translate the three days' time into hours. This is easily done:

Now, based on this, we can set up the equation:

Now, this means that there will be or gallons in the reservoir after three days.

Recall that in general

Now, let's gather our three rates:

Rate 1:

Rate 2:

Rate 3:

Now, we know that the time is a total of hours. Based on our data, we can write:

This is miles per hour, which rounds to .

This problem is a variation on the standard equation . The variable contains all twenty bakers, however, instead of just one. Still, let's start by substituting in our data:

Solving for , we get .

Now, this represents how many dozen cookies the whole group of make per hour. We can find the individual rate by dividing by , which gives us . Notice, however, that the question asks for the number of cookies—not the number of dozens. Therefore, you need to multiply by , which gives you .

This problem is a variation on the standard equation . The variable contains all the workers. Therefore, we could rewrite this as , where is the number of workers and is the individual rate of work. Thus, for our first bit of data, we know:

Solving for , you get

Now, for the actual question, we can fill out the complete equation based on this data:

Solving for , you get .

Begin by setting up the standard equation

However, for our data, we know the distance and the rate only. We do not know the time that it took for the person's return. It is , where is the return time. Thus, we can write:

Solving for , we get:

, which rounds to minutes.

Here, we need to do some unit conversions, knowing that there are in an . We have two different rates, which result in two different equations, which we need to add to get a total time.

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