# the ratio of the number of apples to the number of oranges is 4:6. The ratio of the number of oranges to the number of pears is 8:1. How many apples, Oranges and pears are there if there are 172 fruits.

The answer is 64 apples, 96 oranges, and 12 pears. Let's represent the fruit as following: a - the number of apples, o - the number of oranges, p - the number of pears. [latex]a:o=4:6[/latex] ⇒ [latex] \frac{a}{o}= \frac{4}{6} [/latex] ⇒ [latex]a= \frac{4}{6}o [/latex] [latex]o:p=8:1[/latex] ⇒ [latex] \frac{o}{p}= \frac{8}{1} [/latex] ⇒ [latex]o= 8p [/latex] Therefore: [latex]a= \frac{4}{6}*8p =16/3p [/latex] Now, if [latex]a+o+p=172[/latex], then: [latex] \frac{16}{3}p +8p+p=172[/latex] [latex] \frac{16}{3}p +9p172[/latex] Since [latex]9= \frac{9}{1} = \frac{27}{3} [/latex], 9p can be expressed as 27/3p: [latex] \frac{16}{3}p+ \frac{27}{3}p=172 [/latex] [latex] \frac{43}{3}p =172[/latex] ⇒ [latex]p =172* \frac{3}{43} [/latex] ⇒ p = 12 There are 12 pears. Since o = 8p, o = 96: o = 8 × 12 = 96 There are 96 oranges. Since [latex]a= \frac{16}{3}p[/latex], a = 64: [latex]a= \frac{16}{3} *12=16*4=64[/latex] There are 64 apples. Therefore, there are 64 apples, 96 oranges, and 12 pears.