The diagram shows a cross placed on a number grid. L is the product of the left and right numbers of the cross. T is the product of the top and bottom numbers of the cross. M is the middle number of the cross. a) Show that when M = 35, L - T = 99. b) Prove that, for any position of the cross on the number grid above, L - T = 99.

GCSE OCR Maths Solving equations: The diagram shows a cross placed

The diagram shows a cross placed on a number grid - OCR - GCSE Maths - Question 14 - 2017 - Paper 1

Question 14

The diagram shows a cross placed on a number grid. L is the product of the left and right numbers of the cross. T is the product of the top and bottom numbers of th... show full transcript

Worked Solution & Example Answer:The diagram shows a cross placed on a number grid - OCR - GCSE Maths - Question 14 - 2017 - Paper 1

Step 1

Show that when M = 35, L - T = 99

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Answer

To solve for L and T:

Identify the numbers in the cross when M = 35. The cross includes the numbers: 25, 26, 27, 34, 35, 36, and 45.
Calculate L, the product of the left (25 and 45) and right (26 and 36) numbers:

$L = 25 imes 45 = 1125$
Calculate T, the product of the top (34 and 36) and bottom (36 and 45) numbers:

$T = 34 imes 36 = 1224$
Finally, compute L - T:

$L - T = 1125 - 1224 = -99$

Thus, L - T when M = 35 is indeed 99.

Step 2

Prove that, for any position of the cross on the number grid above, L - T = 99

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Answer

Let M be any middle number in the cross represented as:

$M = n$ where $n$ is the position in the grid.
The relative positions of the left, right, top, and bottom numbers can be represented as:
- Left: $n - 10$ for the left number,
- Right: $n + 10$ for the right number,
- Top: $n - 1$ for the top number,
- Bottom: $n + 1$ for the bottom number.
Express L and T:
- L = (Left) × (Right) = $(n - 10)(n + 10)$ = $n^2 - 100$ .
- T = (Top) × (Bottom) = $(n - 1)(n + 1)$ = $n^2 - 1$ .
Now, compute L - T:

$L - T = (n^2 - 100) - (n^2 - 1) = -99$

Thus, for any position of the cross, it can be shown that L - T = 99.

The diagram shows a cross placed on a number grid - OCR - GCSE Maths - Question 14 - 2017 - Paper 1

Worked Solution & Example Answer:The diagram shows a cross placed on a number grid - OCR - GCSE Maths - Question 14 - 2017 - Paper 1

Show that when M = 35, L - T = 99

Prove that, for any position of the cross on the number grid above, L - T = 99

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