Volume of Right Rectangular Prisms: Fractional Lengths

Volume of Right Rectangular Prisms: Fractional Lengths exercise

• Step by step method.

  Hints

  Start by converting any mixed numbers to improper fractions. For example, $1 \frac{1}{2}$ becomes $\frac{3}{2}$. This will make it easier to multiply the fractions.

  Remember to multiply the fractions in the order given by the formula $V = l \times w \times h$. Multiply the numerators together to get the numerator of the result, and the denominators together to get the denominator of the result.

  After multiplying the fractions, simplify the resulting fraction by dividing the numerator and the denominator by their greatest common divisor. This will give you the simplest form of the volume. Final answers must also contain units too.

  Solution

  The correct order is;

  • Convert the mixed number to an improper fraction.

  $\\$ $1\frac{1}{2} = \frac{3}{2}$

  • Multiply the lengths: $\\$

  $V = \frac{3}{2} \times 2 \times \frac{3}{4}$

  • Simplify the answer fraction to give it in the form the question asks for. $\\$

  $\frac{9}{4} = 2\frac{1}{4}$

  • Don't forget to add the units to your final answer. They should be cubed as volume is in three dimensions.

  $\\$ $2\frac{1}{4}m^3$

• Recognising dimensions.

  Hints

  Carefully look at the diagram and find the labels for the length, width, and height of the cuboid. These labels are usually marked along the edges of the cuboid.

  Take care as the orientation and angle which the prism is viewed at can change which edge may be the length, width or height.

  Make sure you correctly input each dimension into the corresponding gap. The length is typically the longest side, the width is the shorter side on the same face, and the height is the vertical side.

  Remember to multiply the fractions in the order given by the formula $V = l \times w \times h$. Multiply the numerators together to get the numerator of the result, and the denominators together to get the denominator of the result.

  Solution

  The dimensions are:

  Length: $\frac{3}{2}$ metres

  Width: $\frac{5}{4}$ metres

  Height: $\frac{2}{3}$ metres

  Using the formula for finding the volume of a cuboid, you should use the calculation $V = \frac{3}{2} \times \frac{5}{4} \times \frac{2}{3}$. This gives a final volume of $\frac{5}{4}cm^3$.

• Match the cuboid with the corresponding volume.

  Hints

  Before calculating the volume, ensure all dimensions are in fractional form. Convert any mixed numbers to improper fractions to make multiplication easier and more accurate. For example $1 \frac{1}{4} = \frac{5}{4}$.

  Remember the formula for finding the volume of a rectangular prism (cuboid): $V = l \times w \times h$.

  When calculating the volume, make sure to correctly multiply fractions and whole numbers. For example, to multiply $\frac{1}{2}$ by $\frac{3}{4}$, multiply the numerators ($1$ and $3$) to get $3$ and the denominators ($2$ and $4$) to get $8$, so $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$.

  Solution

  The correct pairings for this task are:

  • 1) $V= 1 \frac{1}{2}ft \times 2ft \times \frac{3}{4}ft = 2\frac{1}{4}ft^3 $

  $\\$

  • 2) $V= \frac{5}{6}m \times \frac{7}{8}m \times 1m = \frac{35}{48}m^3$

  $\\$

  • 3) $V= 2in \times \frac{2}{3}in \times 1\frac{1}{4}in = \frac{5}{3}in^3$

  $\\$

  • 4) $V= 1\frac{1}{2}cm \times \frac{4}{3}cm \times \frac{5}{6}cm = \frac{5}{3}cm^3$

• Special cuboids - a cube!

  Hints

  All the dimensions on a cube are the same length! So the width, height and length of the cube are all $\frac{2}{5}cm^3$.

  Put the dimensions into the formula to give us $Vol = \frac{2}{5} \times \frac{2}{5} \times \frac{2}{5}$.

  Don't forget all volumes need units. For this question, they will be in the form $cm^3$ as the cube has dimensions in $cm$.

  Another way of finding the volume of a cube is to use the formula $V = s^3$, where $s$ is the length of the side of the cube!

  Solution

  The correct solution for this question is $\frac{8}{125}cm^3$.

  This is worked out knowing that all the edges of a cube are the same length, so here the length, width and height will all be $\frac{2}{5}cm$. Using the formula, you will get the calculation $Vol = \frac{2}{5} \times \frac{2}{5} \times \frac{2}{5}$ which equals a total volume of $\frac{8}{125}cm^3$.

• Calculating the volume of a cuboid.

  Hints

  Remember the formula for the volume of a cuboid is $V = l \times w \times h$. Ensure you multiply the length, width, and height together to find the volume.

  When multiplying fractions, multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, to multiply $\frac{2}{3}$ and $\frac{4}{5}$, you calculate $\frac{2 \times 4}{3 \times 5} = \frac{8}{15}$.

  After finding the product of the fractions, simplify the result if possible. This means dividing both the numerator and the denominator by their greatest common divisor. For example, $\frac{8}{12}$ can be simplified to $\frac{2}{3}$ because both $8$ and $12$ can be divided by $4$.

  Solution

  The correct solution for this problem is $\frac{1}{4}cm^3$.

  This is worked out using the calculation $V = \frac{3}{4}cm \times \frac{1}{2}cm \times \frac{2}{3}cm$.

  The result to this calculation is $ \frac{6}{24}cm^3$, which simplifies to $\frac{1}{4}cm^3$.

• Finding a missing dimension.

  Hints

  Remember that the volume $V$ of a rectangular prism is calculated using the formula $V = l \times w \times h$, where $l$ is the length, $w$ is the width and $h$ is the height.

  Substitute the given values into the volume formula. You'll have $\frac{3}{10} = l \times \frac{3}{5} \times \frac{6}{4}$. Rearrange the equation to solve for the missing length $l$.

  To isolate $l$, divide the volume by the product of the width and height. This means you need to compute $l = \frac{3}{10} \div (\frac{3}{5} \times \frac{6}{4})$.

  Solution

  The correct answer for the missing length is $\frac{1}{3}cm$.

  This is calculated by substituting the values into the volume formula and then rearranging it to give us the calculation, $l = \frac{3}{10} \div (\frac{3}{5} \times \frac{6}{4})$.

  By completing this you will be given a final solution of $\frac{1}{3}cm$.