This concept sheet offers a number of mental math strategies. The secret to becoming an expert at mental math is practice! The more you practise, the more you develop strategies that work for you. After all, there's often more than one way to complete a calculation in your head.
-
Visualize the operation in a concrete setting
Mathematical calculations are often abstract, but if you think about a real-life context, the calculations become more concrete and easier to visualize.
For example, |6\times 0.25| seems difficult to perform mentally. However, if you think of it as calculating the value of |6| coins worth |25| cents, then you might find the answer easier since you know that it takes |4| coins worth |25| cents to make |\$1.| So, |6\times 0.25=1.50.|
-
Memorize (learn by heart)
To do mental math, you need to know the arithmetic tables. Fortunately, there are a number of tricks you can use to memorize the tables. As well as learning the addition, subtraction, multiplication and division tables, it can also be useful to learn some square numbers like |1,| |4,| |9,| |16,| etc., or powers of |2| like |1,| |2,|4,| |8,|16,| |32,| |64,| etc. The more you build up your knowledge bank, the easier the mental calculations will seem.
-
Estimate
In many situations, the exact answer to a calculation is not needed. A good estimate usually does the trick.
For example, if you buy |3| items at |\$7.95,| |\$12.09| and |\$15.99| and you want to know the total amount of your purchases, you can round the 3 numbers and add them together as follows: |8+12+16.| You get | \$36,| which is very close to the exact answer of |\$36.03.|
-
Round, and remove the zeros from the calculation
There are 3 steps to this process:
1. Round one or more of the numbers used in the operation.
2. Calculate without taking the zeros into account.
3. Readjust the answer.
For example, what is |108 + 34|? Round |108| to |110| and |34| to |30.| The calculation becomes |110+30.| Since |11+3=14,| we know that |110+30=140.| We added |2| units to round |108| to |110| and subtracted |4| units to round |34| to |30.| Therefore, we must subtract |2| and add |4| to |140,| which gives |142.|
-
Break down operations
This involves using the properties of operations (associativity, commutativity and distributivity) to your advantage. It's often easier to perform a calculation in several small, accessible steps than in one single, complicated one.
For example, |35\times 12| is difficult, but |35\times 10+35\times 2| is more accessible. Instead of one multiplication, we multiply 2 times, followed by an addition. There are more operations to do, but at least you can find the answer mentally.
Note: Don't forget to respect the order of operations when breaking a calculation down into several operations.
There's often more than one way to break down a calculation. For example, |35\times 12| could also be broken down as follows: |30\times 12+5\times 12.| It's not important that everyone breaks down their calculation in the same way, but that you break it down in a way that works for you.
The same applies when you have a calculation involving several numbers and decide to use the strategy of rounding to the nearest ten. You don't have to round all the numbers. For example, to calculate |108+34,| you can decide to only round |108| to |110| and not to round the number |34,| so the calculation becomes |110 + 34 -2.|
In the following example, we'll round the numbers, add them together, and then adjust the result at the end.
Find the sum of |139| and |48| mentally.
We can round |139| to the nearest ten, which is |140.|
We can also round |48| to the nearest ten, which is |50.|
To perform this addition, we can ignore the |0|s and add them at the end.
Since |14 + 5| is |19,| then |140 + 50| is |190.|
All that is left to do is to adjust the answer, since the terms of the initial addition were modified.
We added |1| unit to round |139| to |140| and we added |2| units to round |48| to |50.| We must therefore subtract |1| and subtract |2| from |190,| which gives |187.|
Answer: The sum of |139| and |48| is |187.|
Here is a detailed breakdown of the mental calculations performed.||\begin{alignat}{13}139+48&=(139\boldsymbol{\color{#ec0000}{+1}})&&+(48\boldsymbol{\color{#3a9a38}{+2}})&&\boldsymbol{\color{#ec0000}{-}}\boldsymbol{\color{#ec0000}{1}}\boldsymbol{\color{#3a9a38}{-2}}\\[3pt]&=\ \ \ \ \ 140&&+\ \ \ \ \ 50&&-3\\[3pt]&=\ \,14\times10&&+\ \,5\times10&&-3\\[3pt]&=&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!(14+5)\times10&&-3\\[3pt]&=&&\!\boldsymbol{19}0&&-3\\[3pt]&=&&\!187\end{alignat}||
Addition is commutative and associative, which means you can change the order of the calculations without changing the final answer. As a result, we give priority to calculations where no regrouping is required and sums that yield multiples of |10,| or |100,| etc.
For example, if you need to do |94+18+6,| you start by doing |94+6,| which gives |100,| then add |18.|
You can also break down a calculation into simpler, more mentally accessible ones.
For example, instead of doing |67+25,| we do |67+20+5.|
In the following example, we'll round the numbers, subtract them and then adjust the result at the end.
Find the difference between |112| and |90| mentally.
We can round |112| to the nearest ten. This gives |110.|
There's no need to round |90| since it already ends with a |0.|
To subtract, we can ignore the |0|s and add them at the end. Just as |11 - 9| gives |2,| similarly, |110 - 90| gives |20.|
All that remains is to adjust the answer, since the terms of the initial subtraction were modified. We removed |2| units to round |112| to |110.| Therefore, we need to add |2| to |20,| which equals |22.|
Answer: The difference between |112| and |90| is |22.|
Here is a detailed breakdown of the mental calculations performed.||\begin{alignat}{13}112-90&=(112\boldsymbol{\color{#ec0000}{-2}})&&-90&&\boldsymbol{\color{#ec0000}{+}}\boldsymbol{\color{#ec0000}{2}}\\[3pt]&=\ \ \ \ \ 110&&-90&&+2\\[3pt]&=&&20&&+2\\[3pt]&=&&22\end{alignat}||
We can use the properties of associativity and commutativity to help us subtract certain numbers mentally.
We try to prioritize subtractions that don't require regrouping, or borrowing.
We also try to break down a large subtraction into a series of smaller, more accessible subtractions.
For example, instead of doing |253-36,| we perform |253-30-6.|
If you need to perform an operation on numbers that end in |0,| you can ignore them during the calculation. Then, at the end of the operation, add the ignored |0| or move the decimal point.
Find the product of |200 \times 70| mentally.
If we ignore all the |0|s, the multiplication becomes |2 \times 7.| The result of this multiplication is |14.|
We must then add the |0|s that were ignored at the start. Since we ignored |3,| |(2\boldsymbol{\color{#3b87cd}{00}} \times 7\boldsymbol{\color{#3b87cd}0}),| they must be added after the |14.|
Answer: The product of |200 \times 70| is therefore |14\ \boldsymbol{\color{#3b87cd}{000}}.|
Here is a detailed breakdown of the mental calculations performed.||\begin{alignat}{13}200\times70&=2\times100&&\times\ \ \ 7\times10\\[3pt]&=\,(2\times7)&&\times(100\times10)\\[3pt]&=\ \ \ \ 14&&\times\ \ \ \,1\ 000\\[3pt]&=&&\!\!\!\!\!\!14\ 000\end{alignat}||
Find the product of |3.012 \times 200| mentally.
If we ignore the |0|s at the end of the number |200,| the multiplication becomes |3.012 \times 2.| The result of this multiplication is |6.024.|
We must then move the decimal point 2 places to the right since we ignored 2 zeros at the start.||6.024\ \overset{\!\!\times 100}{{\large\longrightarrow}}\ 602.4||
Answer: The product of |3.012 \times 200| is therefore |602.4.|
Here is a detailed breakdown of the mental calculations performed.||\begin{align}3.012\times200&=3.012\times 2\times100\\[3pt]&=6.024\times100\\[3pt]&=602.4\end{align}||
To mentally find the answer to a multiplication with numbers that don't end in |0|s, we can round one of the numbers, perform the calculation, and adjust the final result.
Find the product of |21 \times 6| mentally.
We can round |21| to the nearest ten. The result is |20.| |21 \times 6| becomes |20 \times 6.|
To multiply, we can ignore the |0|s and add them at the end. Since |2 \times 6| equals |12,| then |20 \times 6| equals |120.|
All that is left is to adjust the answer since we modified the terms of the initial multiplication. We were supposed to multiply |21| by |6,| but we only did |20| times |6.| So we need to add |1\times 6=6| to our answer.||120 +6 =126||
Answer: The product of |21 \times 6| is |126.|
Note: The distributive property of multiplication was used.
Here is a detailed breakdown of the mental calculations performed.||\begin{alignat}{13}\boldsymbol{\color{#ec0000}{21}}\times6&=&&\!\!\!\!\!\!\!\!\!\!\!\!\!\boldsymbol{\color{#ec0000}{(20+1)}}\times6\\[3pt]&=20\times6&&+1\times6\\[3pt]&=\ \ \,120&&+\ \ \ 6\\[3pt]&=&&\!126\end{alignat}||
Multiplication is also commutative and associative. So, in a calculation where several multiplications are performed one after the other, you can change the order as you please.
The following tricks can also sometimes be used.
-
Multiplying by |5| is the same as multiplying by |10| and dividing by |2,| since |\boldsymbol{\color{#3b87cd}{\times\,5}}=\boldsymbol{\color{#3b87cd}{\times\dfrac{10}{2}}}.|
-
Multiplying by |20| is the same as multiplying by |2| and multiplying by |10.| The same applies if you multiply by |30,| by |40,| and so on.
-
Multiplying by |25| is the same as multiplying by |100| and dividing by |4,| since |\boldsymbol{\color{#3b87cd}{\times\,25}}=\boldsymbol{\color{#3b87cd}{\times\dfrac{100}{4}}}.|
Sometimes there's more than one way to perform a calculation. It's up to you to find the method that suits you best.
For example, to do |48\times 25,| you can do |48\div 4\times 100| or |12\times (4\times 25)| or |(50-2)\times 25.|
If the 2 numbers in the division end with one or more |0|s, they can be removed to make the division easier. Simply select the smallest number of |0|s from the 2 numbers represented in the division and remove them.
Since the same number of |0|s were removed from both the dividend and the divisor, the correct answer is found directly.
Find the quotient of |200 \div 50| mentally.
Since both of the numbers end with |0|s, we choose the number with the fewest |0|s. There is |1| in |50| and there are |2| in |200.| We therefore remove only one |0| from each number.||200 \div 50\ \Rightarrow\ 20 \div 5||The quotient of this division is |4,| which is also the quotient of the original division.
Answer: The quotient of |200 \div 50| is therefore |4.|
Here is a detailed breakdown of the mental calculations performed.||\dfrac{200}{50}=\dfrac{20\cancel{\times10}}{5\cancel{\times10}} =\dfrac{20}{5} =4||
It is also possible to remove a different number of |0|s from the 2 terms of the division.
-
When more |0|s are removed from the dividend, they must be added back to the final answer.
-
When more |0|s are removed from the divisor, move the decimal point to the left at the end of the answer, according to the number of |0|s removed.
Find the quotient of |720 \div 9| mentally.
By ignoring the |0| in |720,| the division becomes |72 \div 9.| |72| is evenly divisible by |9.| The answer of this division is |8.|
Since we've removed one |0| from the dividend, we need to add it to our answer. |(8 \rightarrow8\boldsymbol{\color{#3a9a38}0}).|
Answer: The quotient of |720 \div 9| is |80.|
Here is a detailed breakdown of the mental calculations performed.||\begin{aligned}\\720\div 9&=\\&=\\&=\\&=\end{aligned} \begin{gather}\qquad\Large\curvearrowleft\\72\times\!10\div 9\\72\div 9\times\!10\\8\times\!10\\80\end{gather}||
Find the quotient of |68.4 \div 200| mentally.
If we ignore all the |0|s, the division becomes |68.4 \div 2.| The answer of this division is |34.2.|
We must then move the decimal point |2| positions to the left, since we removed |2| zeros at the start. ||34.2\ \overset{\!\!\div 100}{\large\longrightarrow}\ 0.342||
Answer: The quotient of |68.4 \div 200| is therefore |0.342.|
Here is a detailed breakdown of the mental calculations performed.||\begin{align}68.4\boldsymbol{\color{#3b87cd}{\div200}}&=(68.4\boldsymbol{\color{#3b87cd}{\div2}})\boldsymbol{\color{#3b87cd}{\div100}}\\[3pt] &=34.2\div100\\[3pt] &=0.342 \end{align}||
Division is neither commutative nor associative. Nevertheless, when there are several divisions to be performed one after the other, they can be performed in any order.||a\div b\div c = a\div c\div b||Furthermore, a complicated division can be broken down into several smaller, easier divisions. For example, instead of dividing by |15,| you can divide by |5| and then by |3,| or vice versa.
The distributive property of division to the left can also be used.||(a\pm b)\div c=a\div c\pm b\div c||Additionally, the following tricks can sometimes be used:
-
Dividing by |5,| is the same as multiplying by |2| and dividing by |10,| since |\boldsymbol{\color{#3b87cd}{\div\,5}}= \boldsymbol{\color{#3b87cd}{\div\dfrac{10}{2}}}=\boldsymbol{\color{#3b87cd}{\times\dfrac{2}{10}}}.|
-
Dividing by |20,| is the same as dividing by |2| and dividing by |10.| The same applies if you divide by |30,| by |40,| and so on.
Calculating the percentage of a certain quantity is very useful in everyday life. Just think about taxes and discounts, which are expressed as percentages. It's very useful to be able to mentally calculate a percentage to get an accurate idea of the amount to be paid.
A percentage is a fraction out of |100.| So, to mentally calculate a percentage, we multiply and then divide. We use the same mental calculation tricks we saw in the sections on multiplication and division. There are other tricks too.
-
|50\ \%| of a number is half. So, simply divide the number by |2.|
-
To calculate |10\ \%| of a number, all you have to do is divide the number by |10.| In other words, move the decimal point one place to the left.
-
From |10\ \%| of a number, we can quickly calculate other percentages of that number. For example, for |30\ \%,| multiply the |10\ \%| you calculated by |3,| while for |5\ \%,| divide the |10\ \%| by |2.|
-
To calculate |15\ \%| of a number, which is a good estimate of sales tax, simply add the |10\ \%| to half of the |10\ \%,| which is |5\ \%.|
-
If an item that costs |\boldsymbol{\$60}| has a |\boldsymbol{30\ \%}| discount, what is the value of the discount?
We start by calculating |10\ \%| of |\$60.| To do this, simply divide by |10| (we remove the |0| from |60|).||10\ \% \text{ of } \$60=\$60\div 10 = \$6||Then, since we want |30\ \%| of the amount, which is |3| times |10\ \%,| we multiply our result by |3.| ||\$6 \times 3 = \$18||
Answer: The value of the discount is |\$18.|
- My bill at the hardware store is |\boldsymbol{\$36,}| plus |\boldsymbol{15\ \%}| in taxes. How much do I have to pay?
We start by calculating |10\ \%| of |\$36.| To do this, simply move the decimal point to the left.||10\ \% \text{ of } \$36.00 = \$3.60||We then calculate |5\ \%| of |\$36.| Since we already know what |10\ \%| of |\$36| is, we can simply divide this |10\ \%| by |2.| ||\begin{align}5\ \% \text{ of } \$36.00 &= (10\ \% \text{ of } \$36.00)\div 2 \\[3pt] &=\$3.60\div 2 \\[3pt] &=\$1.80\end{align}||The total tax payable is the sum of these 2 amounts.||\text{Tax}=3.60+1.80=\$5.40||
Answer: The total amount to pay is |36.00+5.40=\$41.40.|
