Competitive Math Made Easy

Cross multiplication is an important technique for solving equations that contain fractions. In this technique, we multiply both sides of an equation by the denominators of the fractions within it. Doing so will remove the denominators and make the equation easier to solve.  Cross multiplication is based on the principle that for any nonzero values $a$ and $b$,  $  a \cdot \cfrac{b}{a} = b$  In other words, multiplying a fraction by its denominator leaves just its numerator. Example 1 : $ \cfrac {2}{x} = \cfrac{1}{5}$ . First, we multiply both sides by $5$, obtaining $ \cfrac{10}{x} = 1$. Then, we multiply both sides by $x$. This cancels the denominator of the left side, leaving  $x = 10$ Example 2 : Simplify $ \cfrac {a}{1 + \cfrac {4}{5}a} $ Here we must first collapse the denominator into a single term . This can be done using common denominators.      $ \Rightarrow$ $ 1 + \cfrac {4}{5}a = \cfrac {5+4a}{...

Equations are the foundation of mathematics. All forms of math rely on the principle of equality. An equation states that two expressions have the same value. Expressions are what are on either side of the equation. This may seem obvious, but understanding this is essential. Variables, or the letters we see in equations, are values that we do not know. We must manipulate the equation to find the value of the variable(s). Coefficients are the numbers located directly left of variables. The coefficient of a variable multiplies the variable’s value. For example, $5x$ means “five times x.” Linear equations are the building blocks of algebra. An example of a linear equation is  $2x + 3 = 6$. To solve this equation, we must obtain the variable alone on one side and a simplified value on the other. This process is called isolating the variable. It uses principles of inverse operations: subtraction cancels addition, division cancels multiplication, etc. Example 1 : $2x+3=6$  So...

Probability is a quantity that expresses the chance, or likelihood, of an event. It is most helpful to think of probability as a fraction. The literal definition of probability is the chance of occurrence of an event. For example, if a person is standing at the intersection of two roads which direct towards North, South, East, and West. Thus, he has a total of $4$ choices (four different directions) to proceed. Now, if he wished to go towards a particular direction, then the probability of completing his wish is $\frac{1}{4}$ since he can only choose one out of the four directions. Consider another example: A person has two different cars, a Toyota and Honda, which he uses randomly. It can then be said that the probability of using the Toyota is $\frac{1}{2}$ because out of his total of $2$ cars, he can randomly pick $1$ of them. Hence, from the above examples, we can conclude that the probability of an event occurring is $$ = \frac{\text{Number of desired or successful outcomes...

In this blog post, we will study the very  Fundamental Principles of Counting (i) Multiplication  If one operation can be performed in $m$ ways and corresponding to each way of performing the first operation, a second operation can be performed in $n$ ways then the two operations can be performed in $m \cdot n$ ways. In other words, if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \cdot n$ ways of doing both.  Here the different jobs/operations are mutually inclusive. It implies that all the jobs are being done in succession. In this case we use the ' and ' operator to account for all scenarios, and remember ' and ' refers to multiplication. Example :  A student has to select a letter from vowels and another letter from consonants, then in how many ways can he make this selection? Solution : Out of $5$ vowels he can select one vowel in $5$ ways and out of $21$ consonants he can select one consonant i...

Key Facts If a person can finish a job in n days, then the work done by the person in 1 day is $\frac{1}{n}$th of the total job If a person completes $\frac{1}{n}$th of the total job in 1 day, then the time taken by the person to finish the complete job is n days.  Another version of work is pool/tank problems where there is an inlet of water and an outlet as well.   If an inlet fills a tank in n hours, then it fills $\frac{1}{n}$th part of the tank in 1 hour. i.e. work done by it in 1 hour is $\frac{1}{n}$ If an outlet empties a full tank in m hours, then it will empty $\frac{1}{m}$th part of the tank in 1 hour. i.e. work done by it is $-\frac{1}{m}$ The concept is not hard to understand, however the application can be extremely tricky. So it would only help to solve as many problems as you can on this subject. Let's look at some examples. Ex 1. A copy machine can copy a paper in $36$ minutes. If a second copy machine were to be used at the same ...

Units digit of a number is the digit in the one's place of the number. For example, the units digit of $243$ is $3$. In competition math, you might come across occasional problems that ask you to find the units digit of an expression. These problems are more commonly found on Mathcounts Countdown Round. 1. The units digit of any number expressed as a power of $2$ repeats in the cycle $2, 4, 8, 6.$        $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, ....$ etc. 2. The units digit of any number expressed as a power of $3$ repeats in the cycle $3, 9, 7, 1.$        $3^1 = 3, 3^2 = 9, 4^3 = 27, 3^4 = 81, 3^5 = 243, ....$ etc. 3. The units digit of any number expressed as a power of $4$ is $4$ if the power is odd and $6$ if the power is even.        $4^1 = 4, 4^2 = 16, 4^3=64, 4^4=256, ....$ etc. 4. The units digit of any number expressed as a power of $5$ is always $5$.        $5^1...

Key Facts 1. Numbers of the form $\frac{p}{q}, q \neq 0$, where $p$ and $q$ are integers and those that can be expressed in the form of terminating or repeating decimals are called rational numbers .             Ex. $\frac{7}{32} = 0.21875, \frac{8}{15} = 0.5\bar{3}$ are rational numbers. 2. Properties of operations of rational numbers     For any rational numbers $a, b, c,$     (i) Rational numbers are closed under addition, multiplication, and subtraction.          $i.e., (a+b), (a-b),$ and $(a \cdot b)$ are all rational     (ii) Rational numbers follow the commutative law  of addition and multiplication.          $i.e., a + b = b + a$ and $a \cdot b = b \cdot a.$     (iii) Rational numbers follow the associative law of addition and multiplication,          $i.e., (a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = ...

The past 2 weeks that I have been away, were quite the adventure in the Grand Teton and the Yellowstone National Parks. As a family we have always loved spending time outdoors and this was the perfect trip right before the onset of my sophomore year. Getting back to life always seems so hard after a trip like that. Before I digress too much, lets get back to work. Today I would like to review the fundamentals of Circles. Let's first understand the different components of a Circle. Here are the key terms you should know. Radius - Line Segment from the center of the Circle to a point on the circumference. Diameter - Line Segment from one point on the circumference to another point on the circumference that passes through the center of the Circle. Needless to say, the length of the diameter is twice that of the radius. Area - The set of all points contained inside the circumference. Can be found using the formula $ \pi r^2$ where $r$ is the radius. ...

An angle is formed when two rays share an origin. There are many different types of angles formed when two lines/rays/line-segments intersect. The point of origin/intersection is called the  vertex , and the two rays are called the  sides . When we name an angle, we always put the vertex in the middle. For example, in the image on the right, $\angle ABC$ is an angle with vertex at point $B$. Angles: All Different Kinds A $90^{\circ}$ angle is a Right Angle . Lines, segments, or rays that form a right angle are said to be Perpendicular . An angle smaller than $90^{\circ}$ is an Acute Angle . An angle between $90^{\circ}$ and $180^{\circ}$ is an Obtuse Angle . An angle that measures $180^{\circ}$ is a Straight Angle . An angle of more than $180^{\circ}$ is a Reflex Angle . Two angles that add up to $180^{\circ}$ are known as Supplementary Angles . Two angles that add up to $90^{\circ}$ are known as Complementary Angles . When two lines intersect, they ...

Lets look at some basic facts related to sides and angles in triangles. For any triangle, the sum of any two sides will always be larger than the third side. We could also say that the difference of lengths of any two sides must be less than the length of the third side. (Triangle Inequality Theorem) The sum of the three interior angles of a triangle is $180^{\circ}$. The sum of all exterior angles of an n-sided polygon is $360^{\circ}$. The two base angles of an Isosceles triangle are congruent. The sum of all interior angles of an n-sided polygon is $(n − 2) \cdot 180^{\circ}$ . An exterior angle of a triangle is equal to the sum of the two opposite interior angles. (Exterior Angle Theorem) For a triangle, the opposite side of a bigger interior angle is longer than that of a smaller angle, and vice versa. In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. (Pythagorean Theorem) Here are some problems that you would solve using...

Today, we will be looking at problems that ask you to find the number of trailing $0$'s in any factorial. Say for example, we are asked to find the number of trailing zeroes of $101!$ Reminder: $n! = n \cdot (n-1) \cdot (n-2) \cdot ... \cdot 3 \cdot 2 \cdot 1$ Simply put, trailing zeroes are zeroes at the end of the number without any non zero digits to the right of them. Ex. - there are $4$ trailing zeroes in the number $20340000$. Trailing $0$'s are formed when a multiple of $2$ is multiplied by a multiple of $5$. So, all we have to do is count the number of $5$'s and $2$'s in $101!$. Let's start with counting the number of $5$'s. The numbers $5, 10, 15, 20, 25, ... 95, 100$ all contribute one factor of $5$ to the factorial, so we have $20$ factors of $5$. However, some numbers have more than one multiple of $5$. For example, the numbers $25 = 5 \cdot 5$ $50 = 2 \cdot 5 \cdot 5$  $75 = 3 \cdot 5 \cdot 5$ $100 = 4 \cdot 5 \cdot 5$   all have an ex...

The key to solving moving train problems is to understand the distance that the needs to be covered or the relative speeds at which the objects are commuting in the problems. Lets take a look at both of these cases and talk about how to solve these types of questions.  If the train is passing a Stationary object, think Distance  - If the problem asks for the time taken by a moving train to pass a pole or standing man or anything similar that basically is a point, you will need to find the time taken by the train to cover the length of the train itself.   - If the problem asks for the time taken by a moving train to pass a bridge or a tunnel or anything that has a length of itself, then the time taken by the train to pass that object will be the length of the train $+$ the length of the object. If the train is passing a Moving object, think Speed  - If the problem involves $2$ moving objects, you would usually want to calculate the net speed betwee...

Let's take a look at geometric sequences today. In a way, they are similar to Arithmetic Sequences , but just think common ratio instead of common difference. Let's begin by taking a look at the geometric sequence $2, 6, 18, 54...$. The initial term , or the first term is $2$. The common ratio  is defined as the number you multiply to each term to get the next one. In this case, the common ratio would be $3$, because $2 \cdot 3 = 6, 6 \cdot 3 = 18, 18 \cdot 3 = 54$, and so on. Common problems include finding the $n$th term, or finding the sum of such sequences. To find the $n$th term of a geometric sequence, we use the formula: $$a_n = ar^{n-1}$$ Where $a_n =$ the $n$th term in the sequence $a =$ the initial term $r =$ the common ratio Your Turn! $1.$  Find the $7$th term in the sequence $5, 10, 20, 40...$ (Scroll down to the bottom of the page for the answer) Next, let's take a look at how to find the sum of such a sequence. Let's find the sum of the seque...

Age problems is another commonly tested concept on competitive exams. Solving many practice problems is the key to mastering this topic. Interpreting the language of what is given and what is asked of is the biggest challenge you would need to overcome. Lets look at some commonly used phrases throughout these problems.  If the present age is $a$, then $n$ times the present age = $an$. If the present age is $a$, then $n$ years later, age = $a+n$. If the present age is $a$, then age $n$ years ago = $a-n$. The next step would be to organize the given information. What usually works for me is putting a chart together and to keep plugging pieces of data that are given. Let's walk through a few examples to see how this works.  Example 1:  Tim's age is three times his son, Alex's age. After $10$ years, he would only be twice Alex's age. What is Alex's present age?  Solution: Current Age Age $10$ years later Tim $3x$ $3x+10, 2(x+10)$ Alex $x$ $x+10$ ...

Distance and Work problems are another type of problems that you can almost be sure will appear on all math competitions at every level. It is imperative for young olympians to understand them well and get comfortable as early as you can. This post will touch upon the very basics of Speed/Distance/Time and Work problems. There will be future posts where I will discuss different variations and complexity levels within this category. Distance Problems The key in solving Speed/Distance/Time problems is to understand the relationship between each of these concepts. The basic formula to understand here is Distance = Rate $\cdot$ Time $D = R \cdot T$ It is highly recommended to make use of various tools like diagrams and charts to organize the given information leading up to the solution. The formula can be applied to different flavors of such problems. For example, if you know the time and rate a person is traveling on a bus, you can quickly calculate how far he traveled. And if...

What is the Prime Factorization of a Number? The prime factorization of a number is the unique set of prime numbers that, when multiplied, make up the number. For example, the prime factorization of the number $20 = 2^2 \cdot 5$, because both $2$ and $5$ are prime numbers.  Problems that ask you to find the number of factors a given number has are quite common. To solve such problems, first find the prime factorization of your number. Let's take $20$ as an example. As we saw above, $20 = 2^2 \cdot 5$. To find the number of factors that $20$ has, we will add $1$ to each exponent, and then multiply our results. So in this case, we would multiply $(2+1)(1+1) = 3 \cdot 2 = 6$. (Note: the $5$ has an exponent of $1$).  In general, to find the number of factors of a number, add one to each exponent and calculate the product. Less common, but still useful is the sum of the factors of a number. We start the same way as before and find the prime factorization of the number. ...