In this video, we will learn about important mathematical skills. We will learn how to understand and calculate ratios, fractions, and percents, about area and volume of simple shapes, how to use positive and negative indices, and we will learn about simple probability. Understanding the basic math skills is an important building block for all science. And it also helps us in our daily lives. Let’s jump right in.
We will start with ratios, fractions, and percents. Let’s learn what they are and how they are related. Ratios, fractions, and percents are all ways of expressing mathematical data. We’ll start with fractions. A fraction simply defined is a part of a whole. For example, the mass of the sun is made of one-part helium and three parts the element hydrogen. Although this diagram is not realistic because the elements hydrogen and helium are all mixed together, it nevertheless helps us visualize and understand how much of the sun is made of helium and how much of hydrogen.
We can write this information mathematically for helium as one part over or of the four parts of the sun and for hydrogen as three parts over or of the four parts or three parts of the whole. If we simplify this, for helium we get the fraction one over four and for hydrogen three over four. In other words, one-quarter of the sun’s mass comes from helium and three-quarters from hydrogen.
These are called fractions. Fractions have two parts. The number at the top or above the line is the numerator, and the number below the line is the denominator. The denominator gives us the total number of parts, in other words, the whole.
A ratio is the comparison of the size of two numbers. Using our example, we would say that the mass of helium compared to the mass of the whole sun is one part as to four parts, which we write mathematically as one colon four. For hydrogen, there are three parts as compared to the four total parts. And we write this as three as to four or three colon four. Notice that whether written as a fraction or a ratio, the information given to us is the same, just written in a different way.
A percent is a specific type of ratio where the size of a number is compared to the number 100. Percent means the amount in or for every 100. This comes from the Latin phrase “per centum,” which means for every or in each 100. Using our example, imagine that we could split the sun into 100 parts. 25 of the 100 parts would be helium, and the rest would be hydrogen. In other words, 75 of the 100 parts of the mass of the sun come from hydrogen.
We write this mathematically with a percent symbol. 25 percent or 25 for every 100 means the same as one in every four or one-quarter. And 75 percent or 75 for every 100 parts means the same as three in four or three parts of the four parts, which means the same as three-quarters.
Can you see how ratios, fractions, and percents can be interchanged? Let’s write this information in a table. The table shows how the same information can be expressed in three different ways.
Let’s now turn our attention to another mathematical skill. And that is understanding and using area and volume. Sometimes in chemistry and science, it’s necessary to calculate the area of a simple 2D shape or the volume of a 3D shape. We can calculate the area of simple 2D shapes like squares, rectangles, triangles, and circles using formulas.
Area, symbol capital 𝐴, for a square is calculated by finding the length of the sides, which are the same for a square and multiplying them. The area of a rectangle is the length timesed by the breadth or width. The area of a triangle is calculated by taking the width or breadth of the base of the triangle multiplied by the height of the triangle 𝐻 perpendicular to the base divided by two, which is the same as saying a half multiplied by the width times the perpendicular height.
And the area of a circle is equal to 𝜋 multiplied by 𝑟 squared, where 𝑟 is the radius of the circle, or the distance from the middle of the circle to the side, and 𝜋 equals to a value of 3.14. Note that two radii perfectly in line with each other from one side of the circle through the middle to the other side is called a diameter. And diameter 𝑑 equals two times 𝑟. This is important to know, because another mathematical formula that you might come across, 𝐶 equals to two 𝜋𝑟, where 𝐶 is the circumference or length of the entire side of the circle, looks quite similar to the formula for the area of a circle. Both contain 𝜋𝑟 in their formula or equation. Be careful! Circumference and area have similar equations but are very different.
When measuring area, which is really a measure of how much surface there is regardless of whether the lengths, widths, heights, or radii are measured in millimeters, centimeters, or meters, the answer you get will be something units squared, for example, centimeters squared or meters squared. And these areas or surface areas are often calculated when talking about the gas pressure exerted on a surface or the surface area of a catalyst.
Knowing the volume of a 3D object or shape is useful in many parts of science and chemistry. The volume of a cube or cuboid can be calculated by taking the length of a side multiplied by the width or breadth multiplied by the height. For a cube, however, the length, width, and height are all the same value because the sides are squares. So we could say the formula for the volume of a cube is length times length times length.
Volume is a measure of how much space is inside a 3D shape or object. And regardless of how the sides are measured, whether in millimeters, centimeters, or meters, the answer you get from doing a volume calculation is always something units cubed, for example, centimeters cubed or meters cubed.
Let’s do an example of an area and volume calculation. The area of the side of this cube, where the sides measure two centimeters, is length times length. Putting in the length of the two sides of the square that we are looking at, we multiply and get an area of four centimeters squared. Now, if we want the volume of the entire cube, we say length times width times height. But because it is a cube, we could say length times length times length. So two centimeters times two centimeters times two centimeters gives a volume of eight centimeters cubed.
Let’s move on to the next topic, indices. Indices, also called exponents or powers, is the plural of the term index. An index is a number telling us how many times to multiply a base number by itself. In this example, two is the base number and one is the index or exponent or power. The index of one tells us that we need to multiply the number two by itself one time, which is the same as just saying two.
Two to the power of two means we must multiply the base number of two by itself two times, which gives us two multiplied by two, which is equal to four. Two to the exponent three means two multiplied by itself three times, which is two times two times two, which gives an answer of eight.
Can you see that the index is written as a superscript or a small number to the upper-right-hand side of the base number? An important rule to remember is a base number to the exponent or index of zero is always equal to one. When the exponent is a two, we say the base number has been squared. Just like we saw with area, the answer has a unit that has been squared, for example, centimeters squared or meters squared.
And when the exponent or index is a three, we say the base number has been cubed. We saw that volume is measured in units cubed, for example, centimeters cubed or meters cubed.
Sometimes, an index can be positive and sometimes negative. Using this example, two to the exponent two equals two multiplied by two equals four. Because this index is positive, it means that the base number and exponent are actually in a numerator over one, which still gives us an answer of four. However, when the index is negative, this means that the base number and exponent together are both in the denominator under the number one. When we put the base number and index together under the number one, the index becomes positive. If we simplify or solve, we get one over two times two, which is equal to one over four.
Can you see that the answers are very different depending on what the exponent sign is? Positive and negative indices work for and apply to units as well. For example, the density of lead is 11.34 grams per centimeter cubed, which is the same as saying 11.34 grams over centimeters cubed. Because the index for the centimeters cubed is negative, if we want, we can put it in the denominator and the exponent becomes positive. But sometimes it’s easier to write the unit with a negative index. Writing the unit as a numerator and a denominator helps us to figure out what the equation or formula was. We can see that density is equal to mass divided by volume.
Let’s move on to the last math skill in this video, simple probability. Probability refers to how likely an event is to occur, or the chances that something will happen. For example, if we tossed a coin, the probability or likelihood or chances of landing a heads up or a tails up are one out of two for each, or writing it as a fraction, one over two, or as a percent, 50 percent chance of a heads up and 50 percent chance of a tails up.
What about rolling dice? The chances or likelihood or probability of rolling a six is one out of six because there are six sides to the die and so an equal chance of each side landing face up. We can write this is as a fraction of one over six, which is approximately as a percentage 16.7 percent.
Now, in chemistry, we often talk about the probability of molecules or atoms colliding in a reaction vessel and the probability or likelihood that the collision will result in a reaction. Let’s imagine that the orange and blue circles are atoms of different elements in a reaction vessel. These atoms are constantly and randomly moving and sometimes colliding. Now, if a collision occurs, the probability of it being a blue–orange collision is 33 percent. This means that 33 collisions out of 100 collisions are blue–orange collisions. Other collisions will be blue–blue or orange–orange.
Some of these 33 blue–orange collisions out of 100 will result in the blue and orange atoms bouncing off each other. And some will result in a reaction which we call a successful collision. And a new product is formed between the blue and orange elements. If the probability of the blue and orange elements colliding and bouncing off each other is 50 percent and the probability of the blue and orange atoms colliding and successfully reacting is 50 percent, we could calculate the overall probability of blue and orange atoms reacting, which would be 50 percent of 33 for every 100 collisions.
During this calculation, we’d say 50 percent of or multiplied by the 33 collisions per 100, which is the same as saying 50 per 100 or 50 divided by 100 multiplied or of 33, which gives 16.5. But you can’t have 0.5 of a collision, so we will round that up, which gives 17 successful blue–orange collisions out of every 100 collisions in total.
Now it’s time to summarize everything we’ve learnt. In this video, we learnt about fractions, ratios, and percents. We saw that they are three different but related ways of expressing the same information about the parts of a whole. We learnt what area and volume are and the formulas used to calculate the areas and volumes of simple 2D and 3D shapes.
We learnt what indices or exponents are and how they are used to multiply a base number by itself a specific number of times, as well as how to use positive and negative indices. Lastly, we learnt about simple probability, in other words, the likelihood of an event occurring. And we looked at a basic example of a simple probability calculation in chemistry.