中一級數學題

Q1:
Peter is 12 years old and his grandfather is 63 years old now.
(a)After how many years' time will Peter's age be 1/4 of his grandfather's age?
(b)How many years ago was grandfather's age 18 times Peter's?

(a)
Let x be the number of years
(12+x)/(63+x)=1/4
48+4x=63+x
3x=15
x=5
∴ After 5 years' time will Peter's age be 1/4 of his grandfather's age.

(b)
Let y be the number of years
(63+y)/(12+y)=18
63+y=216+18y
-17y=153
y=-9
∴ 9 years ago was grandfather's age 18 times Peter's.



Q2:
A shop sold 129 records by Alan and Anita with a total sales of $12615. Each record by Alan and each record by Anita are sold at $110 and $95 respectively. How many records
(a)by Alan has the shop sold?
(b)by Anita has the shop sold?

(a)
Let x and 129-x be the number of records by Alan and Anita have the shop sold
110x+95(129-x)=12615
110x-95x=360
15x=360
x=24
∴ 24 records by Alan has the shop sold

(b)
129-x
=129-24
=105
∴ 105 records by Anita has the shop sold

1(a) set years to n
12+n/63+n=1/4
4(12+n)=63+n
48+4n=63+n
4n-n=63-48
3n=15
n=5

// 証︰(12+5/63+5=17/68=1/4) [勿抄]

1(b) set years to n
63-n/12-n=18
63-n=18(12-n)
63-n=216-18n
18n-n=216-63
17n=153
n=9

// 証: (63-9/12-9=54/3=18) [勿抄]

2 Set Alan sold to n, Anita sold to a
n+a=129 --1
a=129-n --2
110n+95a=12615 --3
2-->3
110n+95(129-n)=12615
110n+12255-95n=12615
15n=12615-12255
15n=360
n=24
(a) Alan shop sold 24

n+a=129
24+a=129
a= 105

(b)=105

// 証︰(110*24+95*105=2640+9975=12615) [勿抄]