Visualising the Percent of a Quantity

Visualising the Percent of a Quantity exercise

• Find the missing percent using the double number line.

  Hints

  The percent is what the part would be if the whole were $100$.

  This is represented by the proportion, $\frac{\text{Percent}}{100}=\frac{\text{Part}}{\text{Whole}}$.

  Since there are $3$ increments on the top line before $12$ (including zero), divide $12$ by $3$ to find the distance between the increments.

  The percent is the number at the increment that matches with $100$.

  Solution

  The percent is what the part would be if the whole were $100$.

  This is represented by the proportion, $\frac{\text{Percent}}{100}=\frac{\text{Part}}{\text{Whole}}$.

  Using the double number to work out the equivalent ratios, find the increments on the top line.

  Since there are $3$ increments before $12$, divide $12$ by $3$ to find that the distance between the increments is 4.

  Therefore, the top line increments are $0$, $4$, $8$, $12$, $16$, $20$.

  Since $20$ matches with $100$, there are $20$ parts per $100$ or $20\%$.

• Complete the 10x10 grid.

  Hints

  There are $100$ squares in a $10\text{x}10$ grid which represents the whole.

  A percent means parts per $100$. For example, $20\%$ is $20$ parts per $100$.

  This $10\text{x}10$ grid shows $10$ parts per $100$ or $10\%$.

  Make sure when you highlight, you start with the TOP ROW.

  Solution

  Bonsai found $30$ items and $60\%$ are salty.

  We know the whole is $30$, the percent is $60$, but the part is missing.

  The part in this problem is the number of items that are salty.

  We can visualise this situation using a $10\text{x}10$ grid which represents the whole.

  Since there are $100$ squares in the grid, $60$ squares of the grid represent $60$ parts per $100$, or $60\%$.

  We can determine how many items one square gets by dividing the total number of items, $30$, by $100$.

  • $\frac{30}{100}=0.3$

  Each square gets $0.3$ of an item.

  Now we can find the part by adding the items in these $60$ squares together or calculating $60$ times $0.3$.

  • $60(0.3)=18$

  There are $18$ items in the rubbish that are salty.

• Complete the diagram.

  Hints

  There are $10$ evenly sized pieces in the diagram and each piece represents $10\%$.

  A percent means parts per $100$. For example, $70\%$ is $70$ parts per $100$.

  Since each piece represents $10\%$, this diagram represents $20\%$.

  Solution

  Given the percent and part, a diagram can be used to find the whole by splitting it up into evenly sized pieces.

  There are $10$ evenly sized pieces in the diagram and each piece represents $10\%$.

  Shade $5$ pieces in the diagram to represent $50\%$.

  This shaded section represents $22$ items.

  To find the number of items each piece represents, divide $22$ by $5$.

  • $\frac{22}{5}=4.4$

  There are $4.4$ items per piece.

  Counting up all of the items in each piece, we see that all $10$ pieces represents a total of $44$ items.

  • $10(4.4)=44$

• Interpret the diagrams.

  Hints

  In a double number line, the top line represents part and the bottom line represents whole. The percent is what the part would be if the whole were $100$.

  In these diagrams, each piece represents $10\%$.

  The $10\text{x}10$ grids represent the whole and each white box represents the part. The percent is what the part would be if the whole were $100$.

  This diagram represents $30\%$.

  Solution

  Double Number Lines

  • In a double-number line, the top line represents the part and the bottom line represents the whole. The percent is what the part would be if the whole were $100$.
  • The first double number line represents $75\%$ since $75$ on the top line corresponds with $100$ on the bottom line.
  • The second double number line represents $40\%$ since $40$ on the top line corresponds with $100$ on the bottom line.

  Tape Diagrams

  • In these diagrams, each piece represents $10\%$
  • The first diagram represents $50\%$ since there are $5$ shaded blocks.
  • The second diagram represents $80\%$ since there are $8$ shaded blocks.

  $\bf{10\text{x}10}$ Grid

  • The $10\text{x}10$ grids represent the whole and each shaded box represents the part. The percent is what the part would be if the whole were $100$.
  • Since there are $55$ shaded blocks in the first grid, the percent is $55\%$.
  • Since there are $65$ shaded blocks in the second grid, the percent is $65\%$.

• Identify the equation representing a percent.

  Hints

  The equation shows two equivalent ratios. The percent is what the part would be if the whole were $100$.

  One ratio represents percent and the other represents the part to the whole.

  In the following equation, $20$ represents the percent: $\frac{20}{100}=\frac{5}{25}$

  Solution

  The equation shows two equivalent ratios:

  • $\frac{\text{Percent}}{100}=\frac{\text{Part}}{\text{Whole}}$

  The percent is what the part would be if the whole were $100$. This means that percent relates to part and therefore has to line up to the numerator. Since $100$ relates to the whole, it has to line up with the denominator.

• Solve the word problems using a diagram.

  Hints

  For the first word problem, each square in the 10x10 grid represents $0.32$ because $\frac{32\text{ items}}{100\text{ squares}}=0.32$.

  For the second word problem, there are increments of $20$ on the bottom line from $0$ to $100$. If $40$ pupils is $100\%$, divide $40$ by $5$ to fill in the missing increments on the top row. This will help you find the solution.

  For the third word problem, the tape diagram can be split up into $4$ blocks each representing $25\%$.

  Solution

  1. Last year, the Fighting Bees football team won $75\%$ of their games. The team played a total of $32$ games. How many games did they win?

  Use a $10\text{x}10$ grid to help you answer the problem.

  • Since the team won $75\%$, fill in $75$ out of $100$ squares.
  • Each square represents $0.32$ because $\frac{32\text{ items}}{100\text{ squares}}=0.32$.
  • Multiply $0.32$ by $75$ to find the number of games that the football team won.
  • $0.32(75)=24$
  • Therefore, the Fighting Bees football team won $24$ games last year.

  $~$ 2. On the last day of school, $40$ pupils in the Year 6 at Rockville School competed in a bench ball tournament. Mrs. Matthew's class had $12$ pupils who participated. What percentage of Mrs. Matthew's pupils played in the tournament?

  Use a double-number line to help you answer the problem.

  • In the double number line, the top line represents the part and the bottom line represents the whole.
  • In this word problem, the part is the number of pupils that participated from Mrs. Matthew's class, and the whole is the total number of pupils that competed in the tournament.
  • The increments on the bottom line are $0, 20, 40, 60, 80, 100$.
  • The increments on the top line are $0, 8, 16, 24, 32, 40$.
  • Since the percent is what the part would be if the whole were $100$, the percent is $30\%$.

  $~$ 3. Ramona needs a new backpack for school and is looking for a good sale. She searched the internet for a discounted bag and found one for $25\%$ off! Ramona ended up saving $15$ pounds. What was the original price of the backpack?

  Use a tape diagram to help you answer the problem.

  • Split the diagram into $4$ pieces, each representing $15$ pounds and $25\%$.
  • Counting up all the items in the $4$ pieces, $15+15+15+15=60$, means that the original price was $60$ pounds.