“Rigour is to the mathematician what morality is to men.” - André Weil
How are you at mental arithmetic? Do you struggle with multiplication and division?
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Put away your calculator and we’ll soon be working out simple fractions and how to divisions with remainders.
What is a Quotient?
You don’t need to be a mathematician to know that the quotient is the result of a division. A fraction is a number represented by division.
In arithmetic, to get the quotient, you need to divide. The quotient of A divided by B is Q, which means B x Q = A.
You mightn’t always have a quotient. With natural numbers, for example, you only get a quotient of A divided by B if A is a multiple of B.
A quick reminder:
- The natural number above the divide line is called the numerator.
- The number below is called the denominator.
- The denominator shows how many parts you’ll be divided by.
- The numerator shows how many parts you have to divide.
An improper fraction is when the numerator is a multiple of the denominator. Proper fractions are those we tend to see in everyday life such as a quarter of an hour or half a cake.
When you have to add or subtract fractions, such as in mental arithmetic, you need to change how you write each fraction to do it.
In this case, you need to find a common multiple of the denominator. For fractions with a common denominator, you can just simply add or subtract the numerators. The denominator won’t change.
For fractions with different denominators, you need to find a common denominator. Once you’ve done this, you can add and subtract them as you wish.
Now that we’ve quickly covered common denominators, you can start adding and subtracting as you please.
If you need more help, don’t hesitate to get in touch with a private tutor.
How to Work Out the Quotient as a Whole Number
If you’re doing euclidean division or division with a remainder, your answer will be a whole number (integer) and the remainder.
The rational number A is called the dividend and the real number B is called the divisor.
Basically, dividend = quotient x divisor + remainder.
The remainder must always be less than the divisor.
When the remainder of A divided by B is 0, this means that A is divisible by B and that A is a multiple of B and B is a divisor of A.
“The essence of mathematics lies in its freedom.” - Georg Cantor
Ready to solve your first fractions?
Here are some quick tips about divisibility, particularly for mental arithmetic. A positive number is divisible by:
- 2 if it ends in 0, 2, 4, 6, or 8.
- 3 if the sum of the digits is a multiple of 3.
- 4 if the last two digits make a number divisible by 4.
- 9 if the sum of the digits is a multiple of 9.
- 5 if it ends in 0 or 5.
- 10 if it ends in 0.
- 100 if it ends in 00.
You can always practise with any number. By practising, you’ll get better at working out the quotient, the divisor, and the dividend of a fraction.
“Beauty is the first test: there is no permanent place in the world for ugly mathematics.” - G. H. Hardy
How Can You Calculate Decimals
A decimal quotient is when you solve division leaving no remainder and giving an exact decimal value.
Decimal values can let you know:
- Exact values
- Values close to the quotient
- If the divisor features a decimal, you’ll have to change it.
You’re probably wondering how you can do this. Let us show you.
Take a look at 126 divided by 4.
Start from the left of the dividend and count along the digits until you have a number greater than the divisor. In this case, we’ll take 12.
Go along the 4 times tables until you reach the number closest to 12 without going over.
4 x 3 = 12
Write a 3 as the quotient and 0 as the remainder.
Now continue the calculations with the units. Put six into the remainders. You have six units to divide by 4.
Go along the 4 times tables until you reach the number closest to 6 without going over.
6 = (4 x 1) + 2
You can write a 1 in the quotient. 1 x 4 = 4 so place 4 under your six remainder. Now calculate the remainders.
6 - 4 = 2.
You have 2 units remaining.
You can always stop your calculation here with 126 = (4 x 31) + 2, but you can also keep going until you get an exact number.
You’re going to have to add a decimal point to your quotient.
4 is bigger than 2 so we’ll move onto the first decimal and divide 20 by 4.
20/4 = 5
2/4 = 0.5
This gives you .5 at the end of your quotient.
The exact result is 31.5.
Now that you have no numbers to move down from the dividend (126.0), you’ve reached the end of your calculation.
20-20 = 0.
You can check this by multiplying your answer by the divisor. In other cases, you’d have to also add the remainder back on, but since we’ve done decimals, you don’t have to.
31.5 x 4 = 126.
This is correct.
This shows you how useful a decimal quotient can be.
In some cases, the division will never end. You can only get closer to the result the more you do this.
You can also divide a decimal by an integer. To do this, start with the integer. Then, bring the decimal part down into your calculations, too.
If you use a calculator and try dividing something that would usually leave a remainder, the calculator will give you your result as a decimal. For an integer value with a remainder, you can find free long division calculators.
“Mathematics is the most beautiful and most powerful creation of the human spirit.” - Stefan Banach
How to Transform a Quotient into a Product
Now that you know how to calculate quotients as a decimal, you might have worked out that division and multiplication are inextricably linked.
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You need to know your multiplication tables to effectively master division.
Let’s take the example of A, B, and C.
Multiply A by the quotient of B over C. You’ll find that the result is a quotient with the numerator as the product of A x B and the denominator C.
This is, in part, the secret to basic algebra. Just by doing these kinds of exercises, you’ll start to better understand fractions and how they’re used.
Now you should have a better understanding of quotients.
Here’s a quick recap:
- The result of a division is known as the quotient
- Euclidian division results in a whole number with a remainder
- The remainder must always be less than the divisor
Ready to start calculating fractions?
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