If you’re struggling with this week’s questions, this walkthrough should help you get the skills you need.
This week, we’re exploring:
- Multiplying Decimals
- Venn Diagram Notation
- Increasing and Decreasing by a Percentage
- Inverse Proportion with Equations
- Conversion Graphs
Question 1: Multiplying Decimals
There are lots of different methods for multiplying decimals. You were probably shown the column method at primary school, but you may have also come across the grid method and the lattice method (also called Chinese multiplication). At GCSE, it doesn’t matter which method you use.
Evaluate 0.21 × 0.045.
First, it’s important to remember that evaluate just means “work out”. We will look at the grid method to work out this question.
For this method, we split the number into its component parts – each part represents one column of the number.
0.21 = 0.2 + 0.01
0.045 = 0.04 + 0.005
We then draw a 3-by-3 grid and write the parts of 0.21 across the top and the parts of 0.045 down the side.
Now, we simply multiply each row by each column. Be particularly careful with where you place the decimal point – remember 0.04 × 0.2 = 0.008. You can check your answer by making sure the number of decimal places in the question in total is the same as the number of decimal places in the answer. In this case, 0.04 × 0.2 has three decimal places and 0.008 also has three decimal places.
Finally, we find the sum of each of the numbers in the white boxes.
0.21 × 0.045 = 0.008 + 0.0004 + 0.0010 + 0.00005
The biggest flaw in this method is that you have to be very careful with where you place the decimal point. The lattice method eliminates that problem.
Evaluate 4.897 × 0.29.
We will look at the lattice method to work out this question – this is sometimes called Chinese multiplication.
This method takes a little longer to set up, but it makes multiplying decimals much easier.
Since one of the numbers is 4-digits (4.897) and the other is 3-digits (0.029), we start by drawing a 4-by-3 grid and draw a diagonal line in each of the cells:
Now, write the 4-digit value across the top and the other value down the right-hand-side. Make sure you line up the decimal points with the grid lines.
Now, we multiply each row by each column. We put the tens in the top left of the box and the units in the bottom right of the box. Let’s start with the first column:
4 × 0 = 0 so we put 0 in both the top and bottom section.
4 × 2 = 8 so we put 0 in the top section and 8 in the bottom section.
4 × 9 = 36 so we put 3 in the top section and 6 in the bottom section.
Let’s fill in the rest of the grid in the same way.
Now, we add up each diagonal, starting at the bottom. The first diagonal simply adds to 3. However, the second diagonal is 4 + 6 + 1 = 11. So, we carry the tens into the next diagonal and write the units below this diagonal.
Now, we simply do the same for each diagonal.
Finally, we need to determine where the decimal point goes. To do this, we follow the two decimal points horizontally and vertically until they meet. Then, they travel down the diagonal line.
4.897 × 0.29 = 1.42013
Question 2: Venn Diagram Notation
Venn diagram notation is all about knowing the symbols.
Outside of maths, a union is the act of joining together. In Venn diagrams, it means exactly the same thing: the union of A and B is everything that is in A or in B. It’s the two sets joined together. The green section below shows the set A B:
Similarly, intersect means two or more thing that lie across each other. In Venn diagrams, it is the elements that lie across both sets. More visually, it is the point where the two sets cross. The green section below shows the set A B:
An apostrophe (‘) after a set means the complement of the set – this is not the same as a compliment! Two things complement each other if they go well together or make a complete group. It’s this second definition that’s relevant here: the complement of a set is everything that isn’t in that set. The green section below is the set B’:
We can combine these 3 symbols to define any part of a Venn diagram.
Shade the region represented by A’ B.
Start by marking each part of the union. In the diagram below, the areas that represent A’ and the areas that represent B are marked as such:
Since we are looking for the union, we shade any region with a label in as these will either be in A’ or in B (or both).
Shade the region represented by A’ B.
In this case, we start by labelling the areas as before. However, we only shade the region that is labelled both A’ and B:
Question 3: Increasing and Decreasing by a Percentage
The key to increasing or decreasing by a percentage is remembering the original amount is equivalent to 100%. Then, you can multiply or divide the percentage and the original amount by the same thing to find the required percentage – we covered how to do this right back in Week 1 so it may be worth looking back at this week for some practice.
Let’s increase 240 by 17%.
We want to make 17% using small percentages that are easier to find, such as 10%, 5% and 1%. In this case, 17% = 10% + 5% + 1% + 1%
|100% = 240|
|First, divide both sides by 10 to get 10%.||10% = 24|
|Now, divide this by 2 to get 5%.||5% = 12|
|Next, divide 10% by 10 to get 1%.||1% = 2.4|
|Finally, write another 1%||1% = 2.4|
Now, we add together 10%, 5%, 1% and 1% to find 17% and then add this onto the original amount, 240, to increase it by 17%.
Question 4: Inverse Proportion with Equations
We first looked at direct proportion back in week 8. Two (or more) values are directly proportional if they change at the same rate. If one doubles, the other also doubles.
Two values are inversely proportional if they increase at opposite rates. If one doubles, the other halves and vice-versa.
For example, the time taken for some decorators to paint a room. The more decorators there are, the less time it will take. This assumes that the decorators all work at the same rate and that you don’t have so many decorators that they can’t all work!
Let’s say A is inversely proportional to B. Given that , find the value of A when B = 36.
With question like this, we can simply substitute = 36 into this equation.
Question 5: Conversion Graphs
Conversion graphs are a simple way to convert between two proportional amounts – such as two types of currency or hours worked and pay earned.
The graph below shows the exchange rate between US dollars ($) and English pounds (£). Use the graph to convert the following values:
a. Convert £20 to US dollars.
b. Convert $4000 to pounds.
a. To convert £20 to US dollars, we simply draw a horizontal line across at £20 until we meet the line. Then, we draw a line down to the horizontal axis and read the value.
According to this graph, £20 is approximately $22.50.
b. The graph doesn’t go up to $4000 dollars, so we can’t use the graph directly to find this amount. Instead, we convert $40 to pounds and multiply the result by 100 (since 40 × 100 = 4000).
$40 ≈ £35
$4000 ≈ £3500
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