Number Operations Flashcards

All 47 cards in this deck

What does the symbol $\neq$ mean?

Is not equal to.
e.g. $3 + 4 \neq 8$

What does the symbol $\leq$ mean?

Is less than or equal to.
e.g. $x \leq 5$ means $x$ can be 5 or anything smaller.

What does the symbol $\geq$ mean?

Is greater than or equal to.
e.g. $x \geq 2$ means $x$ can be 2 or anything larger.

In an inequality such as $a < b$, which number does the pointed (narrow) end of the symbol face?

The smaller number.
e.g. in $3 < 7$ the point faces the 3.

True or false? $x < 5$ means $x$ could be equal to 5.

False. $x < 5$ means $x$ is strictly less than 5; $x \leq 5$ allows $x = 5$.

True or false? $-2 > -9$.

True. $-2$ is further right on the number line, so it is greater.

What does BIDMAS stand for?

Brackets, Indices, Division, Multiplication, Addition, Subtraction.

What are the steps to work out a calculation using BIDMAS?

e.g. $2 + 3 \times 4^2$

1. Indices: $4^2 = 16$
2. Multiplication: $3 \times 16 = 48$
3. Addition: $2 + 48 = 50$

In BIDMAS, which comes first: multiplication or division?

Neither — they have equal priority, so work from left to right.

True or false? In BIDMAS, all additions must be done before any subtractions.

False. Addition and subtraction have equal priority; work left to right.

Where do roots fit in the order of operations?

With indices (powers) — done after brackets, before multiplying and dividing.

What does the notation $x^{-1}$ mean?

The reciprocal of $x$, that is $\dfrac{1}{x}$.
e.g. $4^{-1} = \frac{1}{4}$

What is the sign of the answer when you multiply or divide two numbers with the same sign?

Positive.
e.g. $-6 \div -2 = 3$

What is the sign of the answer when you multiply or divide two numbers with different signs?

Negative.
e.g. $-4 \times 5 = -20$

What happens when you subtract a negative number?

e.g. $5 - (-3)$

It becomes an addition.
$5 - (-3) = 5 + 3 = 8$

What happens when you add a negative number?

e.g. $4 + (-6)$

It becomes a subtraction.
$4 + (-6) = 4 - 6 = -2$

True or false? $(-3)^2$ and $-3^2$ give the same answer.

False. $(-3)^2 = 9$ but $-3^2 = -9$, because the power applies to the 3 only.

True or false? Multiplying three negative numbers together gives a positive answer.

False. An odd number of negative factors gives a negative answer.

How should a money answer of $2.5$ be written in pounds?

$\pounds 2.50$ — money is written with two decimal places.

What are the steps to work out the change from a purchase?

e.g. buy 3 pens at $\pounds 1.20$ each, paying with $\pounds 5$

1. Find the total cost: $3 \times 1.20 = \pounds 3.60$
2. Subtract from the amount paid: $5 - 3.60 = \pounds 1.40$

True or false? $\pounds 0.7$ is the correct way to write 70p.

False. It should be written $\pounds 0.70$, with two decimal places.

What are the steps to add two decimals using column addition?

e.g. $3.4 + 12.75$

1. Line up the decimal points, filling gaps with zeros: $03.40 + 12.75$
2. Add each column from the right, carrying over
3. Keep the decimal point in line: $16.15$

True or false? To add decimals in columns you line up the last digits on the right.

False. You line up the decimal points, so that digits of the same place value are together.

What is the inverse operation of addition?

Subtraction.
e.g. $7 + 5 = 12$ so $12 - 5 = 7$

How can you check the answer to a subtraction?

Add the answer to the number subtracted; you should get the starting number.
e.g. $9 - 4 = 5$, check $5 + 4 = 9$

When subtracting decimals in a column, what do you do if the top number has fewer decimal places?

e.g. $5.2 - 3.65$

Fill the empty places with zeros: $5.20 - 3.65$

What are the steps to multiply two decimals without a calculator?

e.g. $6.3 \times 2.4$

1. Ignore the decimal points and multiply the digits: $63 \times 24 = 1512$
2. Count the decimal places in the question: $1 + 1 = 2$
3. Put the point so the answer has 2 d.p.: $15.12$

How many decimal places does the answer to a decimal multiplication have?

The same total as the two numbers being multiplied.
e.g. $0.004 \times 0.32$ has $3 + 2 = 5$ d.p.

What are the steps to divide by a decimal without a calculator?

e.g. $8.46 \div 0.15$

1. Multiply both numbers by 100 so the divisor is a whole number: $846 \div 15$
2. Use short division: $846 \div 15 = 56.4$

What is the inverse operation of multiplication?

Division.
e.g. $6 \times 7 = 42$ so $42 \div 7 = 6$

True or false? Multiplying a positive number by a number between 0 and 1 always makes it larger.

False. It makes it smaller, e.g. $8 \times 0.5 = 4$.

What are the steps to use a given calculation to work out a related one?

e.g. given $46 \times 12 = 552$, find $4.6 \times 1.2$

1. Compare each number: $46 \div 10 = 4.6$ and $12 \div 10 = 1.2$
2. Apply the same changes to the answer: $552 \div 10 \div 10$
3. Answer: $5.52$

If one number in a multiplication is multiplied by 10 and the other is divided by 10, what happens to the answer?

It stays the same.
e.g. $23 \times 17 = 230 \times 1.7$

In a division, what happens to the answer if you divide the divisor by 10?

e.g. $391 \div 17 = 23$, so what is $391 \div 1.7$?

The answer is multiplied by 10.
$391 \div 1.7 = 230$

True or false? If $23 \times 17 = 391$, then $2.3 \times 1.7 = 39.1$.

False. Both numbers are 10 times smaller, so the answer is 100 times smaller: $3.91$.

How do the digits of the answer change when you use place value to derive a related calculation?

They do not change — only the position of the decimal point changes.
e.g. $546 \times 43 = 23478$ and $54.6 \times 4.3 = 234.78$

What does it mean to write a systematic list?

Listing outcomes in a fixed order so that none are missed or repeated.

What are the steps to list all possible outcomes systematically?

e.g. all two-digit numbers made from the digits 1, 2, 3 with no repeats

1. Fix the first digit as the smallest and list the rest in order: 12, 13
2. Repeat with each other first digit: 21, 23, then 31, 32
3. Count the list: 6 outcomes

When listing combinations of one item from each of two sets, how do you keep the list systematic?

Keep the first item fixed while you list every choice of the second, then move on.
e.g. AX, AY, BX, BY

True or false? When listing pairs where the order does not matter, AB and BA should both be written down.

False. If order does not matter they are the same pair, so list it once.

How can the product rule be used to check a systematic list is complete?

Multiply the numbers of choices and compare with how many items you listed.
e.g. 3 tops and 2 hats should give $3 \times 2 = 6$ outfits.

State the product rule for counting.

If there are $m$ ways of doing one task and $n$ ways of doing another, there are $m \times n$ ways of doing both.

How do you find the number of ways of making three independent choices?

e.g. 16 shirts, 5 pairs of jeans, 3 jackets

Multiply all three numbers together.
$16 \times 5 \times 3 = 240$ outfits

What are the steps to use the product rule in reverse to find a missing number of options?

e.g. 5 starters, 8 mains and some desserts give 240 meals

1. Multiply the known choices: $5 \times 8 = 40$
2. Divide the total by this: $240 \div 40 = 6$
3. There are 6 desserts

When a problem offers alternative combinations joined by 'or', what do you do with the separate products?

Add them together.
e.g. (starter and main) or (main and dessert) gives $9 \times 15 + 15 \times 8$

True or false? If dividing a total by the known number of choices gives a decimal, the claimed total is impossible.

True. The number of options must be a whole number, so the claim cannot be correct.

How many matches are played if each of $n$ teams plays every other team once?

$\dfrac{n(n-1)}{2}$
e.g. 10 teams give $\frac{10 \times 9}{2} = 45$ matches.