Shortcut 1 :
When two ratios are equal, they are said to be in proportion.
Example :
If a : b = c : d, then a, b, c and d are proportion.
Shortcut 2 :
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
By cross product rule, we have
ad = bc
Shortcut 3 :
Inverse ratios :
b : a is the inverse ratio of a:b and vice versa.
That is, a : b and b : a are the two ratios inverse to each other.
Shortcut 4 :
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) ⋅ (b : a) = (a/b) ⋅ (b/a) = ab/ab = 1.
Shortcut 5 :
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.
earning of the first person = 3x
earning of the second person = 4x
Shortcut 6 :
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, 5 ⋅ 7 = 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.
Shortcut 7 :
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
Shortcut 8 :
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.
Shortcut 9 :
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.
Shortcut 10 :
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.
Shortcut 11 :
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.
Shortcut 12 :
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.
Shortcut 13 :
If one quantity increases or decreases in the ratio a : b, then the new quantity is
= b of the original quantity/a
More clearly, new quantity = (b ⋅ 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 ⋅ 56)/7 = 48 kg.
Hence, David’s new weight is 48 kg.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and Venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
©All rights reserved. onlinemath4all.com
May 23, 22 01:59 AM
Linear vs Exponential Growth - Concept - Examples
May 23, 22 01:42 AM
Exponential vs Linear Growth Worksheet
May 23, 22 01:34 AM
SAT Math Questions on Exponential vs Linear Growth