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Question

我想問f,2MATHS..
about (appications of simultaneous linear equations)

1. The difference between the ages of Nancy and her son is 30. Three years later, her age will be 3 times of her son's. Find the present age of Nancy.

2. If both the numerator and denominator of a fraction increase by 1, the value of the fraction becomes 1/2. If each of them decrease by 1, the value of the fraction becomes 1/4. Find the fraction.

Answer 1

Q1. Let N = nancy present age.
Let M = nancy's son age.

Formula 1 : M+30=N OR M=N-30
Formula 2 : N+3=3*(M+3)

By simpification,
N+3=3M+9

By subsitution,
N+3=3(N-30)+9
N+3=3N-90+9
N+3=3N-81
3N-N=84
2N=84
N=84/2 = 42

Check N=42, M=12
N+3=45
M+3=15
45/15=3

so answer is correct Nancy age is 42, her son is 12.

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Q2 :

Formula 1 : x+1/y+1=1/2
Formula 2 : x-1/y-1=1/4

Solve for y in Formula 1:
x+1=1/2 (y+1)
x+1=1/2y+1/2
x+1/2=1/2y
y=2(x+1/2)
y=2x+1

Subsitute Y into formula 2 :
x-1/(2x+1)-1 = 1/4
x-1/2x=1/4
4x-4 = 2x
2x = 4
x= 4/2 = 2
x=2

y=2x+1 = 5
y=5

so x=2, y=5,

check

2+1/5+1 = 3/6 = 1/2 check
2-1/5-1 = 1/4 check

so verified that x=2, y=5

Please study more in math. Very useful !

alien :)

Answer 2

1. The difference between the ages of Nancy and her son is 30. Three years later, her age will be 3 times of her son's. Find the present age of Nancy.

Let the present age of Nancy be x.
Let the present age of her son be y.

We have

x - y = 30 ... (1)
(x+3) = 3(y + 3) ... (2)

From (1)

x - y = 30

y = x - 30 ... (3)

Put (3) into (2)

(x + 3) = 3(x - 30 + 3)

x + 3 = 3(x - 27)

x + 3 = 3x - 81

2x = 84

x = 42

So the present age of Nancy is 42.

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2. If both the numerator and denominator of a fraction increase by 1, the value of the fraction becomes 1/2. If each of them decrease by 1, the value of the fraction becomes 1/4. Find the fraction.

Let the fraction be x/y (i.e. denominator be x, numerator be y)

We have

(x+1)/(y+1) = 1/2 ... (1)
(x-1)/(y-1) = 1/4 ... (2)

From (1)

(x+1) / (y+1) = 1/2

2(x+1) = y+1

y = 2x + 2 - 1

y = 2x + 1 ... (3)

Put (3) into (2)

(x-1) / (2x+1-1) = 1/4

(x-1)/2x = 1/4

4(x-1) = 2x

4x - 4 = 2x

2x = 4

x = 2

Put x = 2 into (3)

y = 2(2) + 1

y = 5

So the fraction is 2/5.