# Ratio and proportion shortcuts

**Hint 1:****a/b is the ratio of a to b . That is a:b**

**Hint 2:****When two ratios are equal, they are said to be in proportion.**

**Example:****If a:b = c:d, then a,b,c & d are proportion.**

**Hint 3:****Cross product rule in proportion:**

**Product of extremes = Product of means.**

**Example:****Let us consider the proportion a:b = c:d**

**Extremes = a & d, means = b & c**

**Then, as per the cross product rule, we have**

**ad = bc**

**Hint 4:****Inverse ratios:**

**b:a is the inverse ratio of a:b and vice versa.**

**That is, a:b & b:a are the two ratios inverse to each other.**

**Hint 5:****Verification of inverse ratios:**

**If two ratios are inverse to each other, then their product must be 1.**

**That is, a:b & b:a are two ratios inverse to each other.**

**Then, (a:b)X(c:d) = (a/b)X(c/b) = ad/bc = 1**

**Hint 6:****If the ratio of two quantities is given and we want to get the original quantities, we have to multiply both the terms of the ratio by some constant, say “x”.**

**Example:****The ratio of earnings of two persons is 3:4.**

**Then,**

**the earning of the first person = 3x**

**the earning of the second person = 4x**

**Hint 7:****If we want compare any two ratios, first we have to express the given ratios as fractions.**

**Then, we have to make them to be like fractions.**

**That is, we have to convert the fractions to have same denominators.**

**Example:****Compare: 3:5 and 4:7.**

**First, let us write the ratios 3:5 and 4:7 as fractions.**

**That is 3/5 and 4/7.**

**The above two fractions do not have the same denominators. Let us make them to be same.**

**For that, we have to find L.C.M of the denominators (5,7).**

**That is, 5X7 = 35. We have to make each denominator as 35.**

**Then the fractions will be 21/35 and 20/35.**

**Now compare the numerators 21 and 20.**

**21 is greater.**

**So the first fraction is greater.**

**Hence the first ratio 3:5 is greater than 4:7.**

**Hint 8:****If two ratios P:Q and Q:R are given and we want to find the ratio P:Q:R, we have to do the following steps.**

**First find the common tern in the given two ratios P:Q and Q:R. That is Q.**

**In both the ratios try to get the same value for “Q”.**

**After having done the above step, take the values corresponding to P, Q, R in the above ratios and form the ratio P:Q:R.**

**Example:****If P:Q = 2:3 and Q:R = 4:7, find the ratio P:Q:R.**

**In the above two ratios, we find “Q” in common.**

**The value corresponding to Q in the first ratio is 3 and in the second ratio is 4.**

**L.C.M of (3, 4) = 12.**

**So, if multiply the first ratio by 4 and second by 3,**

**we get P:Q = 8:12 and Q:R = 12:21**

**Now we have same value (12) for “Q” in both the ratios.**

**Now the values corresponding to P, Q & R are 8, 12 & 21.**

**Hence the ratio P:Q:R = 8:12:21**

**Hint 9:****If the ratio of speeds of two vehicles in the ratio a:b, then time taken ratio of the two vehicles would be b:a.**

**Example:****The ratio of speeds of two vehicles is 2:3. Then time taken ratio of the two vehicles to cover the same distance would be 3:2.**

**Hint 10:****If the ratio of speeds of two vehicles in the ratio a:b, then the distance covered ratio in the same amount of time would also be a:b.**

**Example:****The ratio of speeds of two vehicles is 2:3. Each vehicle is given one hour time. Then, the distance covered by the two vehicles would be in the ratio 2:3.**

**Hint 11:****If A is twice as good as B, then the work completed ratio of A and B in the same amount of time would be 2:1.**

**Example:****A is twice as good as B and each given 1 hour time. If A completes 2 unit of work in 1 hour, then B will complete 1 unit of work in one hour.**

**Hint 12:****If A is twice as good as B, then the tame taken ratio of A and B to do the same work would be 1:2.**

**Example:****A is twice as good as B and each given the same amount of work to complete. If A takes 1 hour to complete the work, then B will take 2 hours to complete the same work.**

**Hint 13:****If “m” kg of one kind costing $a per kg is mixed with “n” kg of another kind costing $b per kg, then the price of the mixture would be $ (ma+nb)/(m+n) per kg.**

**Hint 14:****If one quantity increased or decreases in the ratio a:b,**

**then the new quantity is = “b” of the original quantity/a**

**More clearly, new quantity = (“b” X original quantity) / a**

**Example:****David weighs 56 kg. If he reduces his weight in the ratio 7:6, find his new weight.**

**New weight = (6 X 56) / 7 = 48 kg.**

**Hence, David’s new weight = 48 kg.**

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