How do you add or subtract two fractions with different denominators?

Rewrite both of them as equivalent fractions over the lowest common denominator . Then add or subtract the numerators only, writing the result over that single denominator.

How do you find the lowest common denominator of two fractions, and what is it for thirds and fifths?

It is the smallest number that both denominators divide into exactly. For 3 and 5 that number is 15, because no smaller number is divisible by both of them.

True or False? Sometimes only one of the two fractions has to be rewritten before they can be added.

True. When one denominator is already a multiple of the other, it is the lowest common denominator itself, so the other fraction is the only one that changes. Adding eighths to quarters is a case of this.

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Fractions (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontMixed Numbers & Improper Fractions
Define mixed number.

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Cards in this collection (21)

Define mixed number.
A mixed number is a number written as a whole number followed by a fraction, such as $3 \frac{3}{4}$ .
The two parts are written side by side with no symbol between them.
True or False?
$3 \frac{3}{4}$ means 3 multiplied by $\frac{3}{4}$ .
False.
The whole number and the fraction are added, so $3 \frac{3}{4}$ means $3 + \frac{3}{4}$ , or 3.75 as a decimal.
Multiplying them would give $\frac{9}{4}$ , which is a completely different number.
Define improper fraction.
An improper fraction, also called a top-heavy fraction, is one whose numerator is at least as big as its denominator, such as $\frac{15}{4}$ .
Its value is therefore 1 or more.
Why can a value like $\frac{15}{4}$ always be written in two different ways?
Because it is greater than 1, so a whole number can be taken out of it with a fraction left over.
That gives the mixed number $3 \frac{3}{4}$ , which stands for exactly the same value as $\frac{15}{4}$ .
How do you convert a mixed number such as $4 \frac{6}{7}$ into an improper fraction?
Multiply the whole number by the denominator and add the numerator, keeping the same denominator.
Here $4 \times 7 + 6 = 34$ , so $4 \frac{6}{7} = \frac{34}{7}$ .
How do you convert an improper fraction such as $\frac{22}{3}$ into a mixed number?
Divide the numerator by the denominator: the whole-number answer becomes the integer part and the remainder goes over the same denominator.
Here $22 \div 3 = 7$ remainder 1, so $\frac{22}{3} = 7 \frac{1}{3}$ .
How do you add or subtract two fractions with different denominators?
Rewrite both of them as equivalent fractions over the lowest common denominator.
Then add or subtract the numerators only, writing the result over that single denominator.
True or False?
$\frac{1}{3} + \frac{1}{5} = \frac{2}{8}$
False.
Adding something to $\frac{1}{3}$ must give an answer bigger than $\frac{1}{3}$ , but $\frac{2}{8}$ is a quarter, which is smaller than both of the fractions being added.
The correct answer is $\frac{8}{15}$ .
How do you find the lowest common denominator of two fractions, and what is it for thirds and fifths?
It is the smallest number that both denominators divide into exactly.
For 3 and 5 that number is 15, because no smaller number is divisible by both of them.
Complete the equivalent fraction that rewrites this one over a denominator of 8.
$\frac{21}{4} = \frac{______}{8}$
The completed fraction is:
$\frac{21}{4} = \frac{42}{8}$
The denominator was multiplied by 2, so the numerator has to be multiplied by 2 as well, or the value would change.
Work out $3 \frac{1}{3} + \frac{3}{5}$ as a single fraction.
Writing $3 \frac{1}{3}$ as $\frac{10}{3}$ and putting both fractions over 15 gives $\frac{50}{15} + \frac{9}{15}$ .
Adding the numerators gives $\frac{59}{15}$ , which is $3 \frac{14}{15}$ as a mixed number.
True or False?
Sometimes only one of the two fractions has to be rewritten before they can be added.
True.
When one denominator is already a multiple of the other, it is the lowest common denominator itself, so the other fraction is the only one that changes.
Adding eighths to quarters is a case of this.
After adding two fractions, what should you always check before writing down the answer?
Whether the numerator and the denominator share a common factor, and cancel it if they do.
For example $\frac{6}{15}$ is not finished, because both parts divide by 3 to give $\frac{2}{5}$ .
How do you multiply two fractions together?
Multiply the numerators together and multiply the denominators together, so $\frac{2}{7} \times \frac{3}{5} = \frac{2 \times 3}{7 \times 5} = \frac{6}{35}$ .
Unlike adding, there is no need for a common denominator.
When cancelling before multiplying, can a numerator cancel with the other fraction's denominator?
Yes, because once the fractions are being multiplied, every numerator is on the top of the whole product and every denominator on the bottom.
So $\frac{3}{25} \times \frac{10}{11}$ cancels a factor of 5 across the two fractions to become $\frac{3}{5} \times \frac{2}{11}$ .
True or False?
A mixed number can be multiplied by a fraction by multiplying the whole numbers and the fraction parts separately.
False.
Every mixed number has to be turned into an improper fraction first, so $3 \frac{3}{5} \times \frac{2}{9}$ becomes $\frac{18}{5} \times \frac{2}{9}$ .
Handling the whole number and the fraction part separately does not give the right answer.
Why is it worth cancelling common factors before multiplying two fractions rather than afterwards?
The numbers you actually multiply stay small, so the arithmetic is easier and the answer arrives already in its simplest form.
Multiplying first gives $\frac{18}{5} \times \frac{2}{9} = \frac{36}{45}$ , which then still has to be cancelled down to $\frac{4}{5}$ .
Define the reciprocal of a fraction.
The reciprocal of a fraction is that fraction turned upside down, so the reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$ .
Every fraction has one, except a fraction whose value is zero.
How do you divide one fraction by another?
Keep the first fraction, change the division into a multiplication, and flip the second fraction, so $\frac{2}{7} \div \frac{3}{4}$ becomes $\frac{2}{7} \times \frac{4}{3}$ .
From that point it is an ordinary multiplication of two fractions.
Complete the result of multiplying a fraction by its own reciprocal.
$\frac{a}{b} \times \frac{b}{a} =______$
The completed result is:
$\frac{a}{b} \times \frac{b}{a} = 1$
Multiplying by a reciprocal undoes multiplying by the fraction, which is why flipping and multiplying does the job of dividing.
Work out $2 \frac{5}{6} \div \frac{7}{9}$ in its simplest form.
The division becomes $\frac{17}{6} \times \frac{9}{7}$ , and cancelling a factor of 3 across the two fractions leaves $\frac{17}{2} \times \frac{3}{7}$ .
Multiplying then gives $\frac{51}{14}$ .

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