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x is directly proportional to y - OCR - GCSE Maths - Question 9 - 2020 - Paper 6

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x is directly proportional to y. y is directly proportional to z. When x = 10, y = 60. When y = 8, z = 1.6. Find a formula for z in terms of x.

Worked Solution & Example Answer:x is directly proportional to y - OCR - GCSE Maths - Question 9 - 2020 - Paper 6

Step 1

Find the proportionality constant for x and y

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Answer

Since x is directly proportional to y, we can express this as:

x=k1yx = k_1 y

where k1k_1 is a constant. Given that when x=10x = 10, y=60y = 60, we can substitute these values to find k1k_1:

10=k1⋅60⇒k1=1060=1610 = k_1 \cdot 60 \Rightarrow k_1 = \frac{10}{60} = \frac{1}{6}.

Step 2

Establish the relationship between y and z

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Answer

Since y is directly proportional to z, we can express this as:

y=k2zy = k_2 z

where k2k_2 is another constant. Given that when y=8y = 8, z=1.6z = 1.6, we can substitute these values to find k2k_2:

8=k2⋅1.6⇒k2=81.6=58 = k_2 \cdot 1.6 \Rightarrow k_2 = \frac{8}{1.6} = 5.

Step 3

Substitute k_1 to find z in terms of x

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Answer

We can now express z in terms of x using the relationships established:

From the first relationship: y=16xy = \frac{1}{6} x

Substituting this into the second equation, we have:

16x=5z\frac{1}{6} x = 5 z

Rearranging for z gives us:

z=15â‹…16x=x30.z = \frac{1}{5} \cdot \frac{1}{6} x = \frac{x}{30}.

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