Which letter of MATH is bigger?

Algebra Level 2

If M + 5 = A 2 − 1 = T 2 + 3 = H − 4 M+5={ A }^{ 2 }-1={ T }^{ 2 }+3=H-4 , Which of the four numbers M M , A A , T T , H H is the largest?

A T H Impossible to determine M

Ronald Chén by Ronald Chén , 32, Guatemala

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3 solutions

Zico Quintina
Jun 30, 2018

We have H = M + 9 H = M + 9 , so clearly H > M H > M .

Consider H = A 2 + 3 ≥ 3 H = A^2 + 3 \ge 3 . If A < 1 A < 1 , this immediately implies H > A H > A ; on the other hand, if A ≥ 1 A \ge 1 , then A 2 ≥ A A^2 \ge A , which means H = A 2 + 3 ≥ A + 3 > A H = A^2 + 3 \ge A + 3 > A .

We can show H > T H > T in a similar manner.

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Note that |A| must be >= 2 since T^2 = A^2 -4, and in order for us to say one parameter is larger than another, the parameters cannot be complex. Ed Gray

Edwin Gray - 2 years, 5 months ago

Let M = 1 M=1 , then

6 = A 2 − 1 6=A^2-1 ⟹ \implies A 2 = 7 A^2=7 ⟹ \implies A = 7 ≈ 2.646 A=\sqrt{7} \approx 2.646

and

6 = T 2 + 3 6=T^2+3 ⟹ \implies T 2 = 3 T^2=3 ⟹ \implies T = 3 ≈ 1.732 T=\sqrt{3} \approx 1.732

and

6 = H − 4 6=H-4 ⟹ \implies H = 10 H=10

Therefore, the largest is H H .

Marvin

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Mahdi Raza
Apr 23, 2020

{ M + 5 = A 2 − 1 ⟹ M = A 2 − 6 T 2 + 3 = A 2 − 1 ⟹ T = A 2 − 4 H − 4 = A 2 − 1 ⟹ H = A 2 + 3 \begin{cases} M+5 = A^2 - 1 \implies M = A^2-6 \\ T^2 + 3 = A^2 - 1 \implies T = \sqrt{A^2 - 4} \\ H-4 = A^2-1 \implies H = A^2 +3 \end{cases}

∴ H \therefore \boxed{H}

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