7. If the age of a father is 6 times the age of his son and the total age of the father and his son is 42, find their ages.
Solution
8. There are only two kinds of fruit for sale, apple and orange. The prices of an apple
and an orange are $4 and $5 respectively. On a certain day, the number of apples
sold is 6 times the number of oranges sold and the sum of money for the fruit sold is
$1 479. Find the total number of fruit sold that day. HKDSE reference 17 Paper 1
Solution

7. The son's age is 6 and his father's is 36.

Let the age of the son be and the age of his father be ,

substitute for in to find the age of the son:

since , substitute the value of to find the age of the father:

8. The total number of fruit sold is 304.

Let oranges be and apples be ,

substitute to find the number of oranges:

now solve for to find the number of apples:

add the two together to find the total number of fruit:

≈ (rounded up because fruit count is a whole number)

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!