Practice Problems (Junior Cert Mathematics): Revision Notes

Practice Problems

Problems:

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Problem 1:

Explanation: Rolling two dice means there are many possible results. We need to think about all the different ways the dice can land. Let's list all the possible outcomes to see how many there are.

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Problem 2:

Explanation: When you spin two spinners, each with different options, the outcomes can be a bit tricky to keep track of. A two-way table helps us see all the possible results clearly.

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Problem 3:

Explanation: When choosing a lunch, there are many combinations possible, depending on the choices for sandwiches, drinks, and desserts. We'll figure out how many different lunches you can make by multiplying the choices together.

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Solutions:

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Problem 1:

1. Step 1: Start with the first die. It has $6$ possible outcomes: $1, 2, 3, 4, 5,$ or $6$. We list these outcomes to make sure we consider all possibilities for the first die.
2. Step 2: For each outcome on the first die, think about what can happen with the second die. It also has $6$ possible outcomes: $1, 2, 3, 4, 5,$ or $6$. By considering each number from the second die for every outcome from the first die, we ensure that we capture all possible combinations.
3. Step 3: List all the combinations:

• If the first die shows $1$, the second die can show $1, 2, 3, 4, 5$, or $6$. That gives us the pairs $(1,1), (1,2), (1,3), (1,4), (1,5)$, and $(1,6)$.

• We repeat this process for every number on the first die, pairing it with each possible number from the second die. The full list of outcomes is:

• $(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)$

• $(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)$

• $(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)$

• $(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)$

• $(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)$

• $(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)$

4. Step 4: Count all the pairs. You should see that there are $36$ $different$ $outcomes$. Why We Do Each Step:

• Step 1 and Step 2: Listing the outcomes for each die helps us to visualise all possible results.
• Step 3: Pairing each outcome from the first die with each outcome from the second ensures we don't miss any combinations.
• Step 4: Counting all the pairs gives us the total number of possible outcomes. Answer: There are $36$ $possible$ $outcomes$ when you roll two dice.

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Problem 2:

5. Step 1: Start by setting up your two-way table. One side of the table will list the colours from Spinner $1$ $(Red, Blue, Green, Yellow)$, and the other side will list the numbers from Spinner $2$ $(1, 2, 3)$. This helps us organise the information clearly and allows us to see all possible outcomes at a glance. The table looks like this:

$Colour \ Number$	$1$	$2$	$3$
$Red$	$R1$	$R2$	$R3$
$Blue$	$B1$	$B2$	$B3$
$Green$	$G1$	$G2$	$G3$
$Yellow$	$Y1$	$Y2$	$Y3$

6. Step 2: Now, let's fill in the table. Each cell in the table represents one outcome. For example:

• If Spinner $1$ lands on $Red$ and Spinner $2$ lands on $1$, the outcome is $R1$.
• If Spinner $1$ lands on $Blue$ and Spinner $2$ lands on $2$, the outcome is $B2$. By filling in each cell, we ensure that every combination of colour and number is accounted for. The completed table should look like this:

$Colour \ Number$	$1$	$2$	$3$
$Red$	$R1$	$R2$	$R3$
$Blue$	$B1$	$B2$	$B3$
$Green$	$G1$	$G2$	$G3$
$Yellow$	$Y1$	$Y2$	$Y3$

7. Step 3: Count all the outcomes in the table. There are $4$ $rows$ and $3$ $columns$, so $4 × 3 = 12$outcomes in total. This multiplication confirms that we have considered all possibilities. Why We Do Each Step:

• Step 1: Setting up the table helps us organise our work and make sure we're considering every possibility.
• Step 2: Filling in each cell ensures we don't miss any combinations.
• Step 3: Counting the outcomes verifies that we've accounted for every possibility. Answer: There are $12$ $possible$ $outcomes$ when you spin both spinners.

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Problem 3:

8. Step 1: Let's break down the choices:

• Sandwiches: You have $4$ options.
• Drinks: You have $3$ options.
• Desserts: You have $3$ options. Listing the number of options helps us see how many choices are available for each part of the meal.

9. Step 2: To find out how many different lunch combinations you can make, multiply the number of options for each part of the meal:

• First, multiply the number of sandwiches by the number of drinks: $4 × 3 = 12$. This gives us the total number of combinations for sandwiches and drinks.
• Then, multiply that result by the number of desserts: $12 × 3 = 36$. This final multiplication includes the dessert choices, giving us the total number of lunch combinations.

10. Step 3: So, by multiplying the choices together, you find that there are $36$ possible $$combinations of a sandwich, drink, and dessert. Why We Do Each Step:

• Step 1: Listing the choices ensures we know what options are available.
• Step 2: Multiplying the options for each part of the meal counts all possible combinations.
• Step 3: The final multiplication gives the total number of different lunch combinations. Answer: There are $36$ $different$ $lunch$ $combinations$ you can make.

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