Grade 5 APSMO Maths Olympiad worksheet featuring problems on symbolic expressions, fraction diagonals, and square piece puzzles.

Problem 1:

Question: If 'a' means '×', 'b' means '÷', 'c' means '-', and 'd' means '+', then what is the value of the expression \(2 \, a \, 60 \, b \, 15 \, c \, 7 \, d \, 2\)?

#### Solution:
First, we replace the symbols with their corresponding operations:
- 'a' means '×'
- 'b' means '÷'
- 'c' means '-'
- 'd' means '+'

The expression becomes:
\[ 2 \times 60 \div 15 - 7 + 2 \]

Now, we solve step by step using the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)):

1. Multiplication and Division (from left to right):
\[ 2 \times 60 = 120 \]
\[ 120 \div 15 = 8 \]

So the expression now is:
\[ 8 - 7 + 2 \]

2. Addition and Subtraction (from left to right):
\[ 8 - 7 = 1 \]
\[ 1 + 2 = 3 \]

Thus, the value of the expression is:
\[ \boxed{3} \]

Problem 2:

Question: If the sum of both diagonals are the same, find the missing value.

The given grid is:
\[
\begin{array}{|c|c|}
\hline
? & \frac{5}{23} \\
\hline
\frac{9}{23} & \frac{3}{23} \\
\hline
\frac{6}{23} & \frac{3}{23} \\
\hline
\end{array}
\]

#### Solution:
We need to ensure that the sum of the elements along both diagonals is the same. Let the missing value be \( x \).

The diagonals are:
1. Top-left to bottom-right: \( x, \frac{9}{23}, \frac{3}{23} \)
2. Top-right to bottom-left: \( \frac{5}{23}, \frac{9}{23}, \frac{6}{23} \)

The sum of the second diagonal is:
\[ \frac{5}{23} + \frac{9}{23} + \frac{6}{23} = \frac{5 + 9 + 6}{23} = \frac{20}{23} \]

For the sums to be equal, the sum of the first diagonal must also be \( \frac{20}{23} \):
\[ x + \frac{9}{23} + \frac{3}{23} = \frac{20}{23} \]

Simplify the left side:
\[ x + \frac{9 + 3}{23} = \frac{20}{23} \]
\[ x + \frac{12}{23} = \frac{20}{23} \]

Isolate \( x \):
\[ x = \frac{20}{23} - \frac{12}{23} \]
\[ x = \frac{20 - 12}{23} \]
\[ x = \frac{8}{23} \]

Thus, the missing value is:
\[ \boxed{\frac{8}{23}} \]

Problem 3:

Question: A square is broken into two pieces. If one piece is as shown above, which is the other piece?

The given piece is:
\[
\begin{array}{ccccccc}
& & & & & & \\
& & & & & & \\
& & & & & & \\
& & & & & & \\
& & & & & & \\
& & & & & & \\
& & & & & & \\
\end{array}
\]

#### Solution:
We need to identify the other piece that, when combined with the given piece, forms a complete square. The given piece has a specific shape, and we need to find the complementary shape from the options.

By examining the options:
- Option (a) has a shape that fits perfectly with the given piece to form a complete square.
- Options (b), (c), and (d) do not fit the given piece to form a complete square.

Thus, the correct answer is:
\[ \boxed{a} \]

Final Answers:

1. \(\boxed{3}\)
2. \(\boxed{\frac{8}{23}}\)
3. \(\boxed{a}\)