CONCEPT 1:
RATIOS:
1. Numbers can be used to make comparisons in day-to-day situations. When comparing any two numbers, sometimes it is necessary to find out how many times one number is greater than the other. In other words, we often need to express one number as a fraction of the other.
2. For example, Akshay scored 76 runs and Harsh scored 19 runs in the finals of the CB series. Then we can say that Akshay scored 4 times as many runs as Harsh. In simple way, by ratio of 4:1
3. Ratios are useful in making comparisons. They represent a relationship between the two quantities of the same unit. One of the values is divided by the other to find the value of one quantity in terms of the other.
4. Ratio of two terms is denoted by a:b which is equal to a/b, where a is the antecedent and b is the consequent.
For example, 5000g of type A rice is mixed with 3 kg of type B rice. The ratio of type A to type B rice can be calculated by dividing the amount of type A rice by the amount of type B rice. To calculate a ratio, the two quantities should be of same unit=> Convert 3 kg to 3000g and then find the ratio.
=> Ratio of type A to type B rice = 5000/3000 = 5/3
The ratio can be represented as 5:3 The fraction 5/3 amounts to the amount of type A rice to the amount of type B rice.
However, the fraction 5/8 represents the amount of type A rice in the mixture.
CONCEPT 2:
SCALING RATIOS:
1. The order of the terms in a ratio is important. => a:b is not the same as b:a
2. The two quantities should be of same unit. For example, 30 marks can be compared with 45 marks but not with Rs.45
3. Ratios are usually reduced to the lowest form for simplicity. Multiplying or dividing the terms in ratio by the same number does not change it
For example, if there are 5000 students in college A, 4000 students in college B and 3500 students in college C, the ratio of students in the three colleges will be 5000:4000:3500 => On simplifying, we get 10:8:7 Both the ratios are equivalent.
4. Ratios can also be expressed in percentages. To express the value of a ratio as a percentage, multiply the ratio by 100.
5. There are many situations where there are more than two quantities and they are not in the same ratio. The ratios can be scaled to find a common ratio.
For example, of the ratio of red marbles to blue marbles is 2:5 and the ratio of blue marbles to yellow marbles is 6:7, then we can find the common ratio using the scaling ratio method.
Red Blue Yellow
2 5
6 7
Blue is common in both the ratios, so find the LCM of 5 and 6 => 30.
The value of 5 corresponds to 30. So, any other value in the same ratio should be multiplied by 6 => The value of 2 corresponds to 12. (2*6=12) So, 2:5 equals to 12:30
Similarly, the value of 6 corresponds to 30. So, any other value in the same ratio should be multiplied by 5=> The value of 7 corresponds to 35 (7*5=35) So, 6:7 equals to 30:35
Therefore, the ratio of red marbles to blue marbles to yellow marbles be 12:30:35
CONCEPT 3:
FINDING THE COMMON RATIO WHEN MORE THAN TWO RATIOS ARE INVOLVED:
1. The formula to find the common ratio when more than two ratios are given is as follows
Let a/b=n1/d1 b/c=n2/d2 c/d=n3/d3 d/e=n4/d4
Then a:b:c:d:e =(n1n2n3n4) : (d1n2n3n4) : (d1d2n3n4) : (d1d2d3n4) : (d1d2d3d4)
Example,
a/b=3/7 b/c=2/11 c/d=4/5 d/e=9/13
a=3*2*4*9 =216 b=7*2*4*9 c=7*11*4*9 d=7*11*5*9 e=7*11*5*13
=> a:b:c:d:e = 216:504:2772:3465:5005
CONCEPT 4:
COMPARISON OF RATIOS:
1. Let us consider two ratios a:b and c:d. Now, a:b is greater than c:d.if a/b > c/d
Multiplying both sides by bd we get ad>bc => a:b is greater than c:d is ad>bc and vice versa.
2. Thus to determine which of the two given ratios a:b and c:d is greater, we compare a*d and b*c where b>0 and d>0
For example, to compare 4:5 and 3:4, we compare 4*4 and 3*5
Since 16>15, 4:5 is greater than 3:4
CONCEPT 5:
PROPERTIES OF RATIOS:
1. When a ratio, say a:b is multiplied with itself, the new ratio formed is a^2:b^2, is known as duplicate ratio. Also a^3:b^3 is called triplicate ratio, a^(1/3) : b^(1/3) => Sub-triplicate ratio
a^(1/2): b^(1/2) => Sub-duplicate ratio. Moreover, b:a is called reciprocal ratio of a:b
2. Consider that we are given two simultaneous equations with three unknown variables (say x, y and z). Although we required a third equation to find all three unknowns, two equations are enough to determine the ratio of the variables, i.e., x:y:z This can be done as follows
p1x+q1y+r1z = 0 and p2x+q2y+r2z = 0
Then x:y:z = (q1r2-q2r1) : (r1p2-r2p1) : (p1q2-p2q1)
3. Multiplying or dividing the same number (say x) to both the numerator and the denominator of a ratio (say a:b) will not change the value of the ratio.
=> a/b = (a*x):(b*x) = (a/x):(b/x)
4. Effect of either adding or subtracting a number (say x) from the numerator and denominator of a ratio a:b
=> If a<b or (a/b)<1 , then for a positive quantity x,
(a+x)/(b+x) > a/b and (a-x)/(b-x) < a/b
=> If a>b or (a/b)>1, then for a positive quantity x
(a+x)/(b+x) < a/b and (a-x)/(b-x) > a/b
5. If a:b = c:d, then a:b = (a+c) : (b+d)
CONCEPT 6:
PROPORTION:
1. The equality of two ratios is called proportion. A proportion is an equation that has two equivalent ratios on either side. In other words, if a/b = c/d then a, b, c and d are in proportion.
This equality of ratios is denoted as a:b :: c:d
2. When a, b, c and d are in proportion, they are called the first, second, third and fourth proportional respectively. a and d are called the extremes and b and c are called the means. When four members are in proportion, the product of extremes is equal to the product of means.
=> if a/b = c/d, then (a*d) =(b*c)
3. The concept of proportion is not restricted to only two ratios. It can be extended to more than two equal ratios. => is a/b = c/d = e/f = g/h then a, b, c, d, e, f, g and h are said to be in proportion.
CONCEPT 7:
CONTINUED PROPORTION:
1. If a/b = b/c, then a, b and c are said to be in continued proportion. In this case, b is called mean proportional and it is also the geometric mean of a and c, b^2= ac
2. Also, in the case of a continued proportion, the ratio of the first and the third proportional is equal to the duplicate ratio of the first and second proportion.
If a/b = b/c, then a/c= a^2/b^2
CONCEPT 8:
PROPERTIES OF PROPORTIONS:
1. If a:b :: c:d or a/b = c/d, then
a/c = b/d [Alternendo law]
d/c = b/a [Invertendo law]
(a+b)/b = (c+d)/d [Componendo law]
(a-b)/b = (c-d)/d [Dividendo law]
(a-b)/(a+b) = (c-d)/(c+d) [Componendo and Dividendo law]
2. If a/b = c/d = e/f = k, then (a+c+e)/(b+d+f) = k
3. Also, if a/b = c/d = e/f = k and p,q and r are real numbers,
then (p*a^n+ q*c^n + r*e^n) / (p*b^n+ q*d^n+ r*f^n) = k^n
4. However, if a/b, c/d, e/f,.... are not equal, then (a+c+e+...) / (b+d+f+...) lies between the highest and lowest of the given fractions.
CONCEPT 9:
RATIO IN INVESTMENT:
1. If P and Q invest amounts a1 and a2 respectively for time period t1 and t2 respectively then the ratio in which the total profit earned will be divided amongst P and Q will be (a1*t1) : (a2*t2)
The unit of measurement of the time period t1 and t2 should be the same.
2. If n people invest amounts a1, a2, a3 and a4 for the time period t1, t2, t3 and t4
then the profit earned will be divided in the ratio (a1*t1) : (a2*t2) : (a3*t3) : (a4*t4)
RATIOS:
1. Numbers can be used to make comparisons in day-to-day situations. When comparing any two numbers, sometimes it is necessary to find out how many times one number is greater than the other. In other words, we often need to express one number as a fraction of the other.
2. For example, Akshay scored 76 runs and Harsh scored 19 runs in the finals of the CB series. Then we can say that Akshay scored 4 times as many runs as Harsh. In simple way, by ratio of 4:1
3. Ratios are useful in making comparisons. They represent a relationship between the two quantities of the same unit. One of the values is divided by the other to find the value of one quantity in terms of the other.
4. Ratio of two terms is denoted by a:b which is equal to a/b, where a is the antecedent and b is the consequent.
For example, 5000g of type A rice is mixed with 3 kg of type B rice. The ratio of type A to type B rice can be calculated by dividing the amount of type A rice by the amount of type B rice. To calculate a ratio, the two quantities should be of same unit=> Convert 3 kg to 3000g and then find the ratio.
=> Ratio of type A to type B rice = 5000/3000 = 5/3
The ratio can be represented as 5:3 The fraction 5/3 amounts to the amount of type A rice to the amount of type B rice.
However, the fraction 5/8 represents the amount of type A rice in the mixture.
CONCEPT 2:
SCALING RATIOS:
1. The order of the terms in a ratio is important. => a:b is not the same as b:a
2. The two quantities should be of same unit. For example, 30 marks can be compared with 45 marks but not with Rs.45
3. Ratios are usually reduced to the lowest form for simplicity. Multiplying or dividing the terms in ratio by the same number does not change it
For example, if there are 5000 students in college A, 4000 students in college B and 3500 students in college C, the ratio of students in the three colleges will be 5000:4000:3500 => On simplifying, we get 10:8:7 Both the ratios are equivalent.
4. Ratios can also be expressed in percentages. To express the value of a ratio as a percentage, multiply the ratio by 100.
5. There are many situations where there are more than two quantities and they are not in the same ratio. The ratios can be scaled to find a common ratio.
For example, of the ratio of red marbles to blue marbles is 2:5 and the ratio of blue marbles to yellow marbles is 6:7, then we can find the common ratio using the scaling ratio method.
Red Blue Yellow
2 5
6 7
Blue is common in both the ratios, so find the LCM of 5 and 6 => 30.
The value of 5 corresponds to 30. So, any other value in the same ratio should be multiplied by 6 => The value of 2 corresponds to 12. (2*6=12) So, 2:5 equals to 12:30
Similarly, the value of 6 corresponds to 30. So, any other value in the same ratio should be multiplied by 5=> The value of 7 corresponds to 35 (7*5=35) So, 6:7 equals to 30:35
Therefore, the ratio of red marbles to blue marbles to yellow marbles be 12:30:35
CONCEPT 3:
FINDING THE COMMON RATIO WHEN MORE THAN TWO RATIOS ARE INVOLVED:
1. The formula to find the common ratio when more than two ratios are given is as follows
Let a/b=n1/d1 b/c=n2/d2 c/d=n3/d3 d/e=n4/d4
Then a:b:c:d:e =(n1n2n3n4) : (d1n2n3n4) : (d1d2n3n4) : (d1d2d3n4) : (d1d2d3d4)
Example,
a/b=3/7 b/c=2/11 c/d=4/5 d/e=9/13
a=3*2*4*9 =216 b=7*2*4*9 c=7*11*4*9 d=7*11*5*9 e=7*11*5*13
=> a:b:c:d:e = 216:504:2772:3465:5005
CONCEPT 4:
COMPARISON OF RATIOS:
1. Let us consider two ratios a:b and c:d. Now, a:b is greater than c:d.if a/b > c/d
Multiplying both sides by bd we get ad>bc => a:b is greater than c:d is ad>bc and vice versa.
2. Thus to determine which of the two given ratios a:b and c:d is greater, we compare a*d and b*c where b>0 and d>0
For example, to compare 4:5 and 3:4, we compare 4*4 and 3*5
Since 16>15, 4:5 is greater than 3:4
CONCEPT 5:
PROPERTIES OF RATIOS:
1. When a ratio, say a:b is multiplied with itself, the new ratio formed is a^2:b^2, is known as duplicate ratio. Also a^3:b^3 is called triplicate ratio, a^(1/3) : b^(1/3) => Sub-triplicate ratio
a^(1/2): b^(1/2) => Sub-duplicate ratio. Moreover, b:a is called reciprocal ratio of a:b
2. Consider that we are given two simultaneous equations with three unknown variables (say x, y and z). Although we required a third equation to find all three unknowns, two equations are enough to determine the ratio of the variables, i.e., x:y:z This can be done as follows
p1x+q1y+r1z = 0 and p2x+q2y+r2z = 0
Then x:y:z = (q1r2-q2r1) : (r1p2-r2p1) : (p1q2-p2q1)
3. Multiplying or dividing the same number (say x) to both the numerator and the denominator of a ratio (say a:b) will not change the value of the ratio.
=> a/b = (a*x):(b*x) = (a/x):(b/x)
4. Effect of either adding or subtracting a number (say x) from the numerator and denominator of a ratio a:b
=> If a<b or (a/b)<1 , then for a positive quantity x,
(a+x)/(b+x) > a/b and (a-x)/(b-x) < a/b
=> If a>b or (a/b)>1, then for a positive quantity x
(a+x)/(b+x) < a/b and (a-x)/(b-x) > a/b
5. If a:b = c:d, then a:b = (a+c) : (b+d)
CONCEPT 6:
PROPORTION:
1. The equality of two ratios is called proportion. A proportion is an equation that has two equivalent ratios on either side. In other words, if a/b = c/d then a, b, c and d are in proportion.
This equality of ratios is denoted as a:b :: c:d
2. When a, b, c and d are in proportion, they are called the first, second, third and fourth proportional respectively. a and d are called the extremes and b and c are called the means. When four members are in proportion, the product of extremes is equal to the product of means.
=> if a/b = c/d, then (a*d) =(b*c)
3. The concept of proportion is not restricted to only two ratios. It can be extended to more than two equal ratios. => is a/b = c/d = e/f = g/h then a, b, c, d, e, f, g and h are said to be in proportion.
CONCEPT 7:
CONTINUED PROPORTION:
1. If a/b = b/c, then a, b and c are said to be in continued proportion. In this case, b is called mean proportional and it is also the geometric mean of a and c, b^2= ac
2. Also, in the case of a continued proportion, the ratio of the first and the third proportional is equal to the duplicate ratio of the first and second proportion.
If a/b = b/c, then a/c= a^2/b^2
CONCEPT 8:
PROPERTIES OF PROPORTIONS:
1. If a:b :: c:d or a/b = c/d, then
a/c = b/d [Alternendo law]
d/c = b/a [Invertendo law]
(a+b)/b = (c+d)/d [Componendo law]
(a-b)/b = (c-d)/d [Dividendo law]
(a-b)/(a+b) = (c-d)/(c+d) [Componendo and Dividendo law]
2. If a/b = c/d = e/f = k, then (a+c+e)/(b+d+f) = k
3. Also, if a/b = c/d = e/f = k and p,q and r are real numbers,
then (p*a^n+ q*c^n + r*e^n) / (p*b^n+ q*d^n+ r*f^n) = k^n
4. However, if a/b, c/d, e/f,.... are not equal, then (a+c+e+...) / (b+d+f+...) lies between the highest and lowest of the given fractions.
CONCEPT 9:
RATIO IN INVESTMENT:
1. If P and Q invest amounts a1 and a2 respectively for time period t1 and t2 respectively then the ratio in which the total profit earned will be divided amongst P and Q will be (a1*t1) : (a2*t2)
The unit of measurement of the time period t1 and t2 should be the same.
2. If n people invest amounts a1, a2, a3 and a4 for the time period t1, t2, t3 and t4
then the profit earned will be divided in the ratio (a1*t1) : (a2*t2) : (a3*t3) : (a4*t4)
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