Multiplication of decimal numbers

Question

I was wondering if anyone knows any good resources to use or tricks to be able to solve these kinds of mental-arithmetic questions(See image below)? Would be really grateful for any help !

Edit: I would have around 10 to 15 seconds for each question and cannot use pen and paper so what I am really looking for are methods to approximate the best answer out of the 3 available.

Answer 1

Are you comfortable with multiplying whole numbers? The following method looks long because I explain every step, but in your head you can just "move the decimal place"

For example $$684.9 \times 1.07$$ You can effectively remove the decimal place by multiplying each number by a power of $10$ to "move the decimal place". So think of it as $$ \left(\frac{10}{10} 684.9 \right) \times \left( \frac{100}{100} 1.07 \right) = \frac{1}{10} \times \frac{1}{100} \times 6849 \times 107 $$ Now you can just multiply $6849 \times 107$ using your usual multiplication algorithm, and move the decimal place of the result $3$ spots to the left, ie multiply by $(1/10)(1/100)$

Answer 2

Break the numbers down and use distributivity (expanding brackets) like in this example: $8* 5.79 = 8*(5 + (7*\frac{1}{10}) + (9*\frac{1}{100})) = (8*5) + (8*7*\frac{1}{10}) + (8*9*\frac{1}{100}) \\ = 40 + (56*\frac{1}{10}) + (72*\frac{1}{100}) = 40 + 5.6 + 0.72$

Answer 3

Since these are multiple choice you can get far just thinking about the shape of the answers. For example, to choose the correct response given $$ 78 \times 51 $$ note that the last digit must be $8$. Then to choose between $4978$ and $3978$ note that the answer should be close to $80 \times 50 = 4000$.

Note that some of the problems require approximation or rounding. The exact answer to the first one is not one of the three choices.

You can think of these strategies as "tricks" or think of them as being comfortable with numbers, not just proficient with the rules. That comfort comes with practice.

Answer 4

For questions 4) and 8) a hint which is not present in the other answers :

Multiplying by $5$ is like dividing by $2$ (doing that, we care only about the digits that are present in the result, not where the decimal point should be places).

For example, the digits of $915 \times 50$ are the same as the digits of $915/2$ which is easier to compute (half of $900 = 450 +$ half of $15 = 7.5$ ; total : $ 457.5$ ; now, as a rough estimate of the result is $1000 \times 50 = 50000$, we must move the decimal point to the right in $457.5$ giving the final answer $45750$.