Ratio and Proportion - ISEE Middle Level Quantitative Reasoning
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The distance between Carson and Miller is 260 miles and is represented by four inches on a map. The distance between Carson and Davis is 104 miles.
Which is the greater quantity?
(a) The distance between Carson and Davis on the map
(b) 
The distance between Carson and Miller is 260 miles and is represented by four inches on a map. The distance between Carson and Davis is 104 miles.
Which is the greater quantity?
(a) The distance between Carson and Davis on the map
(b)
Let 
be the map distance between Carson and Davis. A proportion statement can be set up relating map inches to real miles:
Solve for 
:
Carson and Davis are 
inches apart on the map; 
Let
Solve for
Carson and Davis are
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The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is represented by three and three-fifths inches on a map.
Which is the greater quantity?
(a) The actual distance between Vandalia and Ferrell
(b) 150 miles
The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is represented by three and three-fifths inches on a map.
Which is the greater quantity?
(a) The actual distance between Vandalia and Ferrell
(b) 150 miles
Let 
be the real distance between Vandalia and Ferrell. A proportion statement can be set up relating real miles to map inches:
Solve for 
:
The actual distance between Vandalia and Ferrell is 150 miles.
Let
Solve for
The actual distance between Vandalia and Ferrell is 150 miles.
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Jay has a shelf of books, of which 60% are hardback. The rest are paperback. If 12 are hardback, how many paperbacks are there?
Jay has a shelf of books, of which 60% are hardback. The rest are paperback. If 12 are hardback, how many paperbacks are there?
There are a couple different ways to solve this problem. One way is to set up an equation from the given equation. Essentially, you have to find the total number of books before you can find how many paperbacks. An equation for that could be 
In other works, 12 is 60% of what total amount? (Remember, in equations, we convert percentages to decimals.) Then, you would solve for x to get 20 total books. Once you know the total, you can subtract the number of hardbacks from that to get 8 paperbacks. Another way to solve this equation is to set up a proportion. That would be 
. Then, we could cross multiply to get 
Solving for x would again give you 20 and you would repeat the steps from above to get 8.
There are a couple different ways to solve this problem. One way is to set up an equation from the given equation. Essentially, you have to find the total number of books before you can find how many paperbacks. An equation for that could be
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A given recipe calls for 
cups of butter for every 
cup of flower and 
cups of sugar. If you wish to triple the recipe, how many total cups of ingredients will you need?
A given recipe calls for
This is an easy case of proportions. To triple the recipe, you merely need to triple each of its component parts; therefore, you will have:
cups of butter for every 
cup of flower and 
cups of sugar
Summing these up, you get:
total cups.
This is an easy case of proportions. To triple the recipe, you merely need to triple each of its component parts; therefore, you will have:
Summing these up, you get:
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A witch's brew contains 
newt eyes for every 
lizard tongues. If Aurelia the witch used 
newt eyes in her recipe, how many lizard tongues did she need to use?
A witch's brew contains
To solve this, you need to set up a proportion:
Multiply both sides by 
:
Simplifying, this gives you:
or 
lizard tongues.
To solve this, you need to set up a proportion:
Multiply both sides by
Simplifying, this gives you:
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Isidore could buy 
equally-sized blocks of cheese for 
. How many could he buy for 
?
Isidore could buy
For this problem, set up a proportion:
, where 
represents the number of cheese blocks that Isidore can buy.
To solve this, multiply both sides by 
:
For this problem, set up a proportion:
To solve this, multiply both sides by
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Refer to the above diagram. How many squares should be shaded in if it is desired that the fraction of the squares that are shaded in should be equivalent to the fraction of the circles that are shaded in?
Refer to the above diagram. How many squares should be shaded in if it is desired that the fraction of the squares that are shaded in should be equivalent to the fraction of the circles that are shaded in?
There are four circles, three of which are shaded; there are eight squares. If we let 
be the number of squares to be shaded, then
There are four circles, three of which are shaded; there are eight squares. If we let
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Which is the greater quantity?
(a) 
(b) 
Which is the greater quantity?
(a)
(b)
From a property of proportions, if 
, it follows that 
. Setting 
,
.
From a property of proportions, if
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A witch's brew contains newt eyes for every lizard tongues. If Aurelia the witch used newt eyes in her recipe, how many lizard tongues did she need to use?
A witch's brew contains
To solve this, you need to set up a proportion:
Multiply both sides by :
Simplifying, this gives you:
or lizard tongues.
To solve this, you need to set up a proportion:
Multiply both sides by
Simplifying, this gives you:
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On Katy's facebook page, she has 
friends who are girls, 
friends that are boys, 
friends from her home town, and 100 friends that are from Michigan. What is the ratio of girl to friends from Michigan?
On Katy's facebook page, she has
Katy has 
friends that are girls and 
friends from Michigan, so the ratio is 
. When we divide both numbers by fifty to simplify, the ratio becomes 
, since
and 
.
Katy has
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The distance between Wilsonville and Coleman is 320 miles and is represented by six inches on a map. The distance between Wilsonville and Garrett is 120 miles.
Which is the greater quantity?
(a) The distance between Coleman and Garrett on the map
(b) 
The distance between Wilsonville and Coleman is 320 miles and is represented by six inches on a map. The distance between Wilsonville and Garrett is 120 miles.
Which is the greater quantity?
(a) The distance between Coleman and Garrett on the map
(b)
The closest that Coleman and Garrett can be to each other is 200 miles (if Garrett is between Wilsonville and Coleman); the farthest is 440 miles (if Wilsonville is between Coleman and Garrett).
Call the map distance between Coleman and Garrett 
.
The two extremes of 
can be calculated using proportion statements.
The minimum:
The maximum:
It is therefore unclear whether the map distance is greater than or less than 5 inches.
The closest that Coleman and Garrett can be to each other is 200 miles (if Garrett is between Wilsonville and Coleman); the farthest is 440 miles (if Wilsonville is between Coleman and Garrett).
Call the map distance between Coleman and Garrett
The two extremes of
The minimum:
The maximum:
It is therefore unclear whether the map distance is greater than or less than 5 inches.
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Travis took 45 minutes to drive a total of 40 miles.
Which is the greater quantity?
(a) 55 miles per hour
(b) The average rate at which Travis drove
Travis took 45 minutes to drive a total of 40 miles.
Which is the greater quantity?
(a) 55 miles per hour
(b) The average rate at which Travis drove
Travis drove 40 miles in three-fourths of an hour, so we can divide:
miles per hour,
which is less than 55.
Travis drove 40 miles in three-fourths of an hour, so we can divide:
which is less than 55.
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Mike drove 120 miles to his mother's house. He finished the trip in two hours and 15 minutes. It took him one hour to drive the first 50 miles.
Which is the greater quantity?
(a) Mike's average speed over the first 50 miles
(b) Mike's average speed over the last 70 miles
Mike drove 120 miles to his mother's house. He finished the trip in two hours and 15 minutes. It took him one hour to drive the first 50 miles.
Which is the greater quantity?
(a) Mike's average speed over the first 50 miles
(b) Mike's average speed over the last 70 miles
We can compare miles per hour.
(a) 50 miles over a one-hour period is 50 miles per hour.
(b) 70 miles over a one-and-one-fourth-hour period is
Mike drove an average of 56 miles per hour over the last 70 miles, making (b) greater.
We can compare miles per hour.
(a) 50 miles over a one-hour period is 50 miles per hour.
(b) 70 miles over a one-and-one-fourth-hour period is
Mike drove an average of 56 miles per hour over the last 70 miles, making (b) greater.
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There are fifteen boys in a room and twelve girls. Five more girls enter. What is the ratio of girls to total students in the room after this change occurs?
There are fifteen boys in a room and twelve girls. Five more girls enter. What is the ratio of girls to total students in the room after this change occurs?
After the new girls enter the room, you have 
boys and 
girls. This means that there is a total of 
people in the room. The ratio of girls to boys would be 
.
After the new girls enter the room, you have
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In a bowl of 
pieces of fruit, 
are apples. The rest are kiwis. If the number of apples is doubled, what is the ratio of kiwis to the total number of fruit in the newly enlarged quantity of fruit in the bowl.
In a bowl of
We know that 
of the total 
pieces of fruit are apples. This means that there are:
apples.
Thus far, we know that we must have:
apples
and
kiwis
Now, if we double the apples, we will have:
apples
and
kiwis
This means that the proportion of kiwis to total fruit will be:
or 
, which can be reduced to 
We know that
Thus far, we know that we must have:
and
Now, if we double the apples, we will have:
and
This means that the proportion of kiwis to total fruit will be:
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In a given neighborhood, there are 200 vehicles. Half of these are cars, a quarter are SUVs, five percent are motorcycles, and the remaining amount are trucks. If the number of trucks are doubled, what is the ratio of motorcycles to total vehicles?
In a given neighborhood, there are 200 vehicles. Half of these are cars, a quarter are SUVs, five percent are motorcycles, and the remaining amount are trucks. If the number of trucks are doubled, what is the ratio of motorcycles to total vehicles?
You just need to work this through step-by-step.
We know that half of the vehicles are cars; therefore, 
of them are cars. To find the number of SUVs, multiply 
by 
(a quarter) and get 
SUVs. To find the number of motorcycles, multiply 
by 
to get 
. Finally, there is 
% remaining for trucks; therefore, multiply 
by 
to get 
.
Now, if this is doubled, we have 
trucks. This means that the total number of vehicles is:
vehicles
Therefore, the ratio of motorcycles to total vehicles will be:
Reducing this, you get:
You just need to work this through step-by-step.
We know that half of the vehicles are cars; therefore,
Now, if this is doubled, we have
Therefore, the ratio of motorcycles to total vehicles will be:
Reducing this, you get:
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and 
are positive.
The ratios 20 to 
and 
to 40 are equvalent.
Which is the greater quantity?
(a) 
(b) 
The ratios 20 to
Which is the greater quantity?
(a)
(b)
The ratios 20 to 
and 
to 40 are equvalent, so
By the cross-product property,
Without any futher information, however, it cannot be determined which of 
and 
is the greater. For example, 
and 
fits the condition, as does the reverse case.
The ratios 20 to
By the cross-product property,
Without any futher information, however, it cannot be determined which of
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and are positive.
The ratios 125 to and to 125 are equvalent.
Which is the greater quantity?
The ratios 125 to
Which is the greater quantity?
The ratios 125 to and to 125 are equvalent, so
By the cross-product property,
Without any futher information, however, it cannot be determined which of and is the greater. For example, 
and 
fits the condition, as does the reverse case.
The ratios 125 to
By the cross-product property,
Without any futher information, however, it cannot be determined which of
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. Which of the following must be equivalent to the ratio
?
(a) 
(b) 
(c) 
(a)
(b)
(c)
Two ratios are equivalent if and only if their cross products are equal. Set 
equal to each choice in turn and find their cross products:
(a) 
The cross products are equal, so regardless of the value of 
, the ratios are equivalent.
(b) 
The cross products are equal if and only if 
, so the ratios are not equivalent.
(c) 
The cross products are equal if and only if 
, so the ratios are not equivalent.
The correct response is (a) only.
Two ratios are equivalent if and only if their cross products are equal. Set
(a)
The cross products are equal, so regardless of the value of
(b)
The cross products are equal if and only if
(c)
The cross products are equal if and only if
The correct response is (a) only.
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and 
are positive. Which is the greater quantity?
(a) 
(b) 
(a)
(b)
The cross products of two equivalent fractions are themselves equivalent, so if
then
Multiply by 6:
Since 
, it follows that 
, and by substitution,
.
The cross products of two equivalent fractions are themselves equivalent, so if
then
Multiply by 6:
Since
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