Multiply and divide positive fractions and decimals

Mixed number

\(\displaystyle 1\frac{2}{3}\)

Which improper fraction has the same value as the mixed number?

A fraction of a fraction

Multiplication by a whole number can describe repeated addition

Multiplication by a fraction describes taking that fraction of a quantity

Half of three fifths is three tenths: splitting each fifth into two equal parts makes tenths

Multiply the numerators and multiply the denominators, then simplify

Here the denominators are non-zero. Unlike addition, multiplication does not need a common denominator

Multiply two proper fractions

Calculate exactly. Give a fraction in simplest form

Both operands are positive, so the result is positive

Cancel the common factor 3 from the second numerator and first denominator

Multiply the numerators and denominators

Check that the final numerator and denominator share no factor greater than one

Find a fraction of a fraction

Simplify common factors before multiplying

A common factor anywhere in the product numerator and product denominator can be divided out before multiplying

Divide the factors three and nine by three, and the factors two and four by two

This is the same result as multiplying first and simplifying six thirty-sixths. Cancellation can keep the intermediate numbers smaller

Cancel common factors, not digits and not two factors that are both in the numerator

Simplify the product

Convert whole and mixed numbers before multiplying

Write a whole number over one. Convert a mixed number to an improper fraction

Then use the same fraction multiplication rule. Do not multiply the whole and fractional parts separately as if they were one product

Multiply a whole number by a fraction

Calculate exactly. Give a fraction in simplest form

Both operands are positive, so the result is positive

Multiply the numerators and denominators

Check that the final numerator and denominator share no factor greater than one

Write the whole number over one

Multiply with a mixed number

Calculate exactly. Give a fraction in simplest form

Both operands are positive, so the result is positive

Write the first mixed number as an improper fraction, keeping its sign

Cancel the common factor 5 from the first numerator and second denominator

Cancel the common factor 3 from the second numerator and first denominator

Multiply the numerators and denominators

Check that the final numerator and denominator share no factor greater than one

Convert, multiply and simplify

Division asks how many groups fit

Dividing three quarters by one quarter asks how many quarter-sized groups fit into three quarters

The inverse multiplication checks the number of groups

The reciprocal of a non-zero fraction is the fraction with numerator and denominator exchanged. A fraction times its reciprocal is one

Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal. Keep the first fraction unchanged

The divisor must be non-zero, so its numerator cannot be zero

Divide by a proper fraction

Calculate exactly. Give a fraction in simplest form

Both operands are positive, so the result is positive

Keep the dividend and multiply by the reciprocal of the divisor

Cancel the common factor 2 from the first numerator and second denominator

Multiply the numerators and denominators

The reciprocal belongs to the divisor only. The original dividend stays unchanged

Check that the final numerator and denominator share no factor greater than one

Multiply by the divisor reciprocal

Convert before finding a reciprocal

The reciprocal of a whole number is one over that number

Convert a mixed-number divisor to one improper fraction before taking its reciprocal

A mixed number is a sum of its whole and fractional parts. Its reciprocal is not found by inverting only the fraction part

Divide a fraction into equal shares

Divide by a mixed number

Calculate exactly. Give a fraction in simplest form

Both operands are positive, so the result is positive

Write the second mixed number as an improper fraction, keeping its sign

Keep the dividend and multiply by the reciprocal of the divisor

Cancel the common factor 3 from the first numerator and second denominator

Cancel the common factor 2 from the second numerator and first denominator

Multiply the numerators and denominators

The reciprocal belongs to the divisor only. The original dividend stays unchanged

Check that the final numerator and denominator share no factor greater than one

Convert both mixed numbers

Division

\(\displaystyle \frac{2}{3}\div\frac{4}{7}\)

Which multiplication is equivalent to the division?

Multiply decimals by counting place-value scales

Two tenths times three tenths is six hundredths

Multiply the numbers as whole numbers, then divide by the combined power-of-ten scale

A factor with two decimal places and a factor with one decimal place give a scale of one thousand

A trailing zero may be removed from an exact decimal without changing its value. These tasks ask for an exact result, not a specified measurement precision

Multiply a decimal greater than one

Calculate exactly. Give a decimal or whole-number answer without rounding

The first factor has 1 decimal place and the second has 1

Multiply the whole-number numerators, then account for all 2 decimal places in the scale

This result is exact, so no rounding is needed

Combine the two decimal-place scales

Include leading zeros in the product

Calculate exactly. Give a decimal or whole-number answer without rounding

The first factor has 2 decimal places and the second has 1

Multiply the whole-number numerators, then account for all 3 decimal places in the scale

This result is exact, so no rounding is needed

Keep all the places needed for the exact product

Scale both the dividend and divisor

A quotient is unchanged when both operands are multiplied by the same non-zero number

Choose a power of ten that makes the divisor a whole number. You may make both operands whole numbers to simplify the calculation

If the divisor is multiplied by one hundred, the dividend must also be multiplied by one hundred

Check division by multiplying the result by the original divisor

All decimal division questions here have exact terminating answers. Do not round them

Use a hundredfold scale for both values

Calculate exactly. Give a decimal or whole-number answer without rounding

Multiply both the dividend and divisor by 100 so both become whole numbers

Scaling both by the same non-zero factor preserves the quotient

Check by multiplying the quotient by the original divisor

This result is exact, so no rounding is needed

Keep the quotient unchanged

A quotient can be less than one

Calculate exactly. Give a decimal or whole-number answer without rounding

Multiply both the dividend and divisor by 10 so both become whole numbers

Scaling both by the same non-zero factor preserves the quotient

Check by multiplying the quotient by the original divisor

This result is exact, so no rounding is needed

Divide a smaller positive quantity by a larger one

Compare the factor or divisor with one

For a positive starting value, multiplying by a number less than one makes the result smaller. Dividing by a positive number less than one makes the result larger

Multiplying or dividing by one leaves the value unchanged. For a number greater than one, multiplication increases the value and division decreases it

These size checks can expose a misplaced decimal point or an incorrect reciprocal

Keep the starting value at three. Move the scale setting to compare a factor of one half with a factor of two

Which statement holds when the selected factor is one half?

Check by the inverse and distinguish zero cases

Check a division by multiplying the quotient by the original divisor

Zero times any number is zero. Zero divided by a non-zero number is zero

Division by zero is undefined. No number multiplied by zero can give a non-zero dividend

Zero divided by zero also has no unique quotient, because every number multiplied by zero gives zero

The generated practice in this topic uses positive non-zero operands

Find a fraction of a measured length

A ribbon has the length shown in metres. One half of the ribbon is used. Find the length used in metres

Taking one half of a length means multiplying the length by one half

The two factors each have one decimal place, giving a scale of one hundred

The product is smaller than the original length because the positive factor is less than one

The length used is one point two metres

Take a decimal fraction of a length

Count equal portions

To count equal portions, divide the total amount by the amount in one portion

A one-and-a-half litre container is filled into portions of one quarter litre each

There are six portions. The answer is a count, not a volume

Find the number of equal portions

Keep the operation and scale visible

Convert mixed numbers before operating. Multiply fraction numerators and denominators, cancelling common factors and simplifying the final answer

For division, keep the dividend and multiply by the reciprocal of the divisor

For decimal multiplication, account for the combined place-value scale. For decimal division, scale both operands equally

Check a quotient by multiplying it by the original divisor, and compare the factor or divisor with one to check the size

Keep exact results exact. State the requested unit when the answer represents a measurement