RatioandProportion

RatioComparing ratiosProportionRateConverting ratesAverage rate of speed

### Ratio

A ratio is a compare of 2 numbers. We typically separate the two numbers in the ratio with a colon (:). Mean we want to compose the proportion of 8 and also 12.We deserve to write this as 8:12 or together a portion 8/12, and also we to speak the ratio is eight come twelve.

You are watching: A statement that two ratios are equal

Examples:

Jeannine has a bag through 3 videocassettes, 4 marbles, 7 books, and also 1 orange.

1) What is the ratio of books to marbles?Expressed together a fraction, through the molecule equal come the first quantity and also the denominator same to the second, the answer would certainly be 7/4.Two other ways of creating the ratio are 7 to 4, and 7:4.

2) What is the proportion of videocassettes to the total number of items in the bag?There space 3 videocassettes, and also 3+4+7+1=15 item total.The answer have the right to be expressed as 3/15, 3 come 15, or 3:15.

### Comparing Ratios

To to compare ratios, write them as fractions. The ratios space equal if they space equal once written together fractions.

Example:

Are the ratios 3 to 4 and also 6:8 equal?The ratios space equal if 3/4=6/8.These room equal if your cross commodities are equal; that is, if 3×8=4×6. Since both of these products equal 24, the prize is yes, the ratios are equal.

Remember to it is in careful! stimulate matters!A ratio of 1:7 is no the exact same as a proportion of 7:1.

Examples:

Are the ratios 7:1 and also 4:81 equal? No!7/1>1, but 4/81

Are 7:14 and also 36:72 equal?Notice that 7/14 and also 36/72 space both same to 1/2, so the two ratios are equal.

### Proportion

A relationship is an equation v a proportion on each side. That is a explain that 2 ratios space equal.3/4=6/8 is an instance of a proportion.

When one of the 4 numbers in a ratio is unknown, cross assets may be offered to uncover the unknown number. This is called solving the proportion. Question marks or letter are frequently used in location of the unknown number.

Example:

Solve because that n: 1/2= n/4.Using cross commodities we watch that 2× n =1×4=4, for this reason 2× n =4. Dividing both political parties by 2, n =4÷2 so that n =2.

### Rate

A rate is a proportion that expresses how long that takes to execute something, such as traveling a specific distance. To walk 3 kilometers in one hour is come walk in ~ the price of 3 km/h. The portion expressing a rate has actually units of street in the numerator and units the time in the denominator.Problems entailing rates frequently involve setup two ratios equal to every other and solving for an unknown quantity, the is, fixing a proportion.

Example:

Juan runs 4 kilometres in 30 minutes. At that rate, how far could he run in 45 minutes?Give the unknown amount the name n. In this case, n is the number of km Juan can run in 45 minute at the given rate. We know that running 4 km in 30 minutes is the exact same as running n km in 45 minutes; the is, the prices are the same. So we have the proportion4km/30min= n km/45min, or 4/30= n/45.Finding the cross commodities and setup them equal, we acquire 30× n =4×45, or30× n =180. Separating both sides by 30, we discover that n =180÷30=6 and also the price is 6 km.

### Converting rates

We compare rates just as we compare ratios, by cross multiplying. When comparing rates, always check to view which devices of measurement space being used. For instance, 3 kilometers every hour is really different native 3 meters every hour!3 kilometers/hour=3 kilometers/hour×1000 meters/1 kilometer=3000 meters/hourbecause 1 kilometer equates to 1000 meters; we "cancel" the kilometers in converting to the units of meters.

Important:

One the the most helpful tips in solving any type of math or science difficulty is to constantly write the end the units when multiplying, dividing, or convert from one unit to another.

Example:

If Juan runs 4 km in 30 minutes, how plenty of hours will it take it him to operation 1 km?Be cautious not to confuse the systems of measurement. When Juan"s price of speed is offered in regards to minutes, the inquiry is make in regards to hours. Only among these units may be provided in setup up a proportion. To convert to hours, multiply4 km/30 minutes×60 minutes/1 hour=8 km/1 hourNow, let n it is in the number of hours it takes Juan to run 1 km. Then running 8 kilometres in 1 hour is the exact same as running 1 kilometres in n hours. Solving the proportion,8 km/1 hour=1 km/n hours, we have 8× n =1, for this reason n =1/8.

### Average rate of Speed

The median rate of speed for a trip is the full distance traveled split by the complete time that the trip.

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Example:

A dog to walk 8 kilometres at 4 kilometres per hour, then chases a rabbit for 2 kilometres at 20 kilometres per hour. What is the dog"s average rate of speed for the street he traveled?The full distance travel is 8+2=10 km.Now us must figure the total time he was traveling.For the first part of the trip, the walked because that 8÷4=2 hours. That chased the hare for 2÷20=0.1 hour. The full time for the expedition is 2+0.1=2.1 hours.The mean rate of speed for his trip is 10/2.1=100/21 kilometers per hour.