Problem 29, Upper Primary: Australian Mathematics Competition 2023

Let's discuss a problem from the AMC 2023 Upper Primary: Problem 29 which revolves around basic algebra.

Problem

Yifan has a construction set consisting of red, blue and yellow rods. All rods of the same colour are the same length, but differently coloured rods are different lengths. She wants to make quadrilaterals using these rods.

• When she uses two red, one blue and one yellow rods, the perimeter of the quadrilateral is 36 cm.
• When she uses two blue, one red and one yellow rod, the perimeter is 35 cm.
• When she uses two yellow, one blue and one red rod, the perimeter is 33 cm.

What number do you get when you multiply the lengths of one red rod, one blue rod and one yellow rod?

Solution

Let's consider the Red rods to be \(R\), the Blue rods to be \(B\) and the Yellow rods to be \(Y\).
According to the 1st point :\(2R + B + Y = 36\)............... (1)

According to the 2nd point: \(2B + R + Y = 35\)..............(2)

According to the 3rd point: \(2Y + B + R = 33\)..............(3)

We have to find the individual values of \(R, Y, B\).

Let's add all the three equation \((1), (2)\) & \((3)\), we get :

\(4R + 4B + 4Y = 36 + 35 + 33\)

\(\therefore\) \(4R + 4B + 4Y = 104\)

\(\therefore\) \(R + B + Y = \frac {104}{4}\)

\(\therefore\) \(R + B + Y = 26\)......................(4)

Now, subtracting (4) from (1):
\(2R + B + Y - R - B - Y= 36 - 26\)
\(R = 10\)

Now, subtracting (4) from (2):
\(2B + R + Y - R - B - Y= 35 - 26 \)
\(B = 9\).

Now implementing the values of \(R\) and \(B\) in the (1) equation we will get:
\(2R + B + Y = 36\)
\(\therefore\) \( 20 + 9 + Y = 36\)
\(\therefore\) \(Y = 36 - 29 = 7\).

So, when we multiply the length of \(R, Y , B\) we get : \( 7 \times 10 \times 9 = 630\).
So the answer is \(630\).


What is AMC (Australian Mathematics Competition)?

The Australian Mathematics Competition (AMC) is one of Australia's largest and oldest annual mathematics competitions, aimed at fostering interest and excellence in mathematics among students.

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