Ration and Proportion math for competitive exam ~ অনলাইন শিক্ষা

A Ratio is a comparison of two similar quantities obtained by dividing one quantity by the other. The quantities to be compared should be of the same unit. Two similar quantities when compared, will not give any unit. The ratio is written with " : " symbol. The ratio of two quantities are written as, a : b. It can also be written as $\frac{a}{b}$ $\frac{a}{b}$ $\frac{a}{b}$
$\frac{a}{b}$ $\frac{a}{b}$ $\frac{a}{b}$ .
When two ratios are equal, they are called Proportions. The proportion is written as a : b = c : d. The cross product of the two ratios can also be equalized a x d = b x c. If three values in a proportion are known, then we can easily find the fourth value. For the proportion to be used, the ratios must be equal.
A ratio is a relationship between two numbers of the same kind. Usually expressed as A : B or $\frac{A}{B}$ $\frac{A}{B}$ $\frac{A}{B}$
$\frac{A}{B}$ $\frac{A}{B}$ $\frac{A}{B}$ or by the phrase "A to B", where, 'A' indicates the quotient of two mathematical expressions and 'B' is the relationship that can be in quantity, amount or size between two or more things.
It is a statement of how two numbers are compared. It is the comparison of the size of one number to the size of another number.

\frac{a}{b}

\frac{a}{b}

\frac{a}{b}

\frac{a}{b}

Find the unknown valu 2 : x = 3 : 9
First, I convert the colon-based odds-notation ratios to fractional form:
.
Then I solve the proportion:
9(2) = x(3)
18 = 3x 6 = x

If twelve inches correspond to 30.48 centimeters, how many centimeters are there in thirty inches?

I will set up my ratios with "inches" on top, and will use "c" to stand for the number of centimeters for which they've asked me.

12c = (30)(30.48) 12c = 914.4 c = 76.2
Thirty inches corresponds to 76.2 cm.

How many feet per second are equivalent to 60 mph?

For this, I will need conversion factors, which are just ratios. If you're doing this kind of problem, then you should have access (in your text or a handout, for instance) to basic conversion factors. If not, then your instructor is probably expecting that you have these factors memorized. I'll set everything up in a long multiplication so that the units cancel:

Then the answer is 88 feet per second.

Find the unknown value in the proportion:  2 : x = 3 : 9.

2 : x = 3 : 9
First, I convert the colon-based odds-notation ratios to fractional form:
.
Then I solve the proportion:
9(2) = x(3)
18 = 3x 6 = x
#Tom is selling apples and oranges. The ratio of apples to oranges in his cart is 3:2. If he has 12 oranges, how many apples does he have?
(A) 2               (B) 3               (C) 8                (D) 18             (E) 30

Solution:
A ratio is basically a fraction that has been reduced as much as possible. In this problem the ratio 3:2, can be represented as 3/2. One way to solve this problem is to set up a simple equation:

Notice I placed the 12, the number of oranges, in the denominator. We have to make sure that the number 12 corresponds to 2, the oranges in the ratio. Solving for x, we get 18 (D).

An even quicker way is to notice that we have (x6) the oranges (from 2 we go to 12) so we just have to (x6) the apples in the ratio: 3 x 6 =18.

Now’s let’s try the same question but with a spin:
# Tom is selling apples and oranges. The ratio of apples to oranges in his cart is 3:2. If he has a total of 30 fruits, how many apples does he have?
(A) 2               (B) 3               (C) 12             (D) 18             (E) 30

Solution:
This question, while essentially the same, is the one that gives students a lot more trouble. The problem is combining two concepts: ratio and total. To do so simply add the ratios. We have 3:2 so the total is 5.
One way to solve the problem is to set up the table. Tables are great both from a teacher’s and beginner’s standpoint. In this case, I get to show you a nice, tidy way of solving the problem and you have an easy way both to conceptualize and solve the problem.
However, once you become used to tables, in the interest of time, learn to solve a ratio without one (I’ll show you how to do so in a second!).

Apples Oranges Total

Ratio 3 2 5

M(x)

Actual ? ? 30

What do we multiply the total ratio by to get the actual total? (x6).
So in the middle row in the total column we can place a 6.

Apples Oranges Total

Ratio 3 2 5

M(x) 6 6 6

Actual 18 12 30

Notice the (M)x, which stands for multiply (you can dispense with the M, I just didn’t want anyone thinking there is this random variable x floating around).
Now we multiply the apples and oranges by 6 to get 18 and 12, respectively.
Remember the faster way I mentioned?

Add the ratio

Figure out the x6

Multiply 3 x 6

18 (D). Also remember not to mix up apples and oranges. A classic trick on the GRE is they reverse the order.

Practice Questions
Okay, you’ve got the hang of ratios. Now you want something a little more challenging. Voila – it’s the ratio challenge workout.
The problems below appear superficially similar. Yet, the math behind each is different. Nonetheless, each of them deals with ratios, and will put your ability to the test. The problems tend to get progressively harder. So if you struggled at the end, don’t despair. As long as you can nail the first three, you’re doing pretty well on ratios.
Good luck!

1. A jewel necklace contains only emeralds, rubies, and diamonds. If the ratio of emeralds to diamonds is 2:7 and the ratio of diamonds to rubies is 3:2, then which of the following could not be the number of jewels on the necklace?
(A) 41
(B) 81
(C) 82
(D) 123
(E) 205

2. A tiara is studded with a mixture of gems. The ratio of sapphires to emeralds is 3:1. If 6 emeralds are added, the tiara will contain an equal number of sapphires and emeralds. How many emeralds must be added to the original tiara so that the ratio between emeralds and sapphires is 3:1?
(A) 9
(B) 12
(C) 18
(D) 24
(E) 27

3. An imperial scepter is mounted with diamonds, rubies, emeralds, and sapphires in the ratio of w : x : y : z. If w, x, y, and z are distinct single digit primes, then how many gems could be on the imperial scepter if it only contains the gems listed above?
(A) 19
(B) 58
(C) 85
(D) 97
(E) Cannot be determined from the information provided above.

4. (Okay – enough with the jewels!) Marty has a coin collection, which consists of only New World and Old World coins, in a ratio of 3:1. Marty’s friend Kyle swaps 28 of his Old World Coins for 28 of Martin’s New World coins. If Marty now has as many Old World coins as he does New World coins, how many coins did Marty originally have in his collection?
(A) 28
(B) 84
(C) 112
(D) 130
(E) 390

5. (Okay, back to jewel necklaces). Marty has a necklace that contains a total of 36 rhinestones, zirconium, and obsidian “gems.” If the ratio of obsidian to zirconium is 2:5, then which of the following could not be the number of rhinestones in Marty’s necklace?
(A) 8
(B) 12
(C) 15
(D) 22
(E) 29

Answers:
1. B
2. D
3. C
4. C
5. B

	Apples	Oranges	Total
Ratio	3	2	5
M(x)
Actual	?	?	30

	Apples	Oranges	Total
Ratio	3	2	5
M(x)	6	6	6
Actual	18	12	30

#: If the ratio of y to x is equal to 3 and the sum of y and x is 80, what is the value of y?
(A) -10 (B) -2 (C) 5 (D) 20 (E) 60
Translating "the ratio of y to x is equal to 3" into an equation yields: y/x = 3 Translating "the sum of y and x is 80" into an equation yields: y + x = 80 Solving the first equation for y gives: y = 3x. Substituting this into the second equation yields 3x + x = 80 4x = 80 x = 20 Hence, y = 3x = 3(20) = 60. The answer is (E).
In many word problems, as one quantity increases (decreases), another quantity also increases (decreases). This type of problem can be solved by setting up a direct proportion.

#: If Biff can shape 3 surfboards in 50 minutes, how many surfboards can he shape in 5 hours?
(A) 16 (B) 17 (C) 18 (D) 19 (E) 20
As time increases so does the number of shaped surfboards. Hence, we set up a direct proportion. First, convert 5 hours into minutes: 5 hours = 5 x 60 minutes = 300 minutes. Next, let x be the number of surfboards shaped in 5 hours. Finally, forming the proportion yields 3/50 = x/300 3(300)/50 = x 18 = x The answer is (C).
If one quantity increases (or decreases) while another quantity decreases (or increases), the quantities are said to be inversely proportional. The statement "y is inversely proportional to x" is written as y = k/x where k is a constant.
Multiplying both sides of y = k/x by x yields yx = k Hence, in an inverse proportion, the product of the two quantities is constant. Therefore, instead of setting ratios equal, we set products equal.
In many word problems, as one quantity increases (decreases), another quantity decreases (increases). This type of problem can be solved by setting up a product of terms.

#: If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?
(A) 3hrs (B) 3 1/2hrs (C) 4 2/3hrs (D) 5hrs (E) 6 1/3hrs
As the number of workers increases, the amount time required to assemble the car decreases. Hence, we set the products of the terms equal. Let x be the time it takes the 12 workers to assemble the car. Forming the equation yields 7(8) = 12x 56/12 = x 4 2/3 = x The answer is (C).

$495

$510

$480

$375

$360

The correct choice is (A) and the correct answer is $495.

$\times$

$\frac{4}{3}$

\times

\times

\frac{8}{540}

\frac{8}{540}

\frac{8}{540}

\frac{8}{540}

\frac{8}{540}

\frac{8}{540}

\frac{1}{67.5}

\frac{1}{67.5}

\frac{1}{67.5}

\frac{1}{67.5}

\frac{1}{67.5}

\frac{7 x + 2}{4}

\frac{7 x + 2}{4}

\frac{7 x + 2}{4}

\frac{7 x + 2}{4}

\frac{7 x + 2}{4}

\frac{(x + 5)}{2}

\frac{(x + 5)}{2}

\frac{(x + 5)}{2}

\frac{(x + 5)}{2}

\frac{(x + 5)}{2}

\frac{8}{5}

অনলাইন শিক্ষা

বুধবার, ৭ ডিসেম্বর, ২০১৬

Ration and Proportion math for competitive exam

$\times$

$\frac{4}{3}$

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