# Fruits

(I) bought three apples, two oranges, four pears and five bananas, which in total cost £3.73.
(i) I bought twenty one apples, no oranges, eight pears and fifteen bananas, which in total cost £7.57.
(i) I bought one apple, eighteen oranges, fourteen pears and sixteen bananas, which in total cost £18.40.
(i) I bought five apples, thirteen oranges, eleven pears and two bananas, which in total cost £12.68.

How much did each pear cost?

p =  0.35

### Step-by-step explanation:

3a+2o+4p+5b=3.73
21a+8p+15b=7.57
1a+18o+14p+16b=18.40
5a+13o+11p+2b=12.68

3·a+2·o+4·p+5·b=3.73
21·a+8·p+15·b=7.57
1·a+18·o+14·p+16·b=18.40
5·a+13·o+11·p+2·b=12.68

3a+5b+2o+4p = 3.73
21a+15b+8p = 7.57
a+16b+18o+14p = 18.4
5a+2b+13o+11p = 12.68

Pivot: Row 1 ↔ Row 2
21a+15b+8p = 7.57
3a+5b+2o+4p = 3.73
a+16b+18o+14p = 18.4
5a+2b+13o+11p = 12.68

Row 2 - 3/21 · Row 1 → Row 2
21a+15b+8p = 7.57
2.8571b+2o+2.8571p = 2.6486
a+16b+18o+14p = 18.4
5a+2b+13o+11p = 12.68

Row 3 - 1/21 · Row 1 → Row 3
21a+15b+8p = 7.57
2.8571b+2o+2.8571p = 2.6486
15.2857b+18o+13.619p = 18.0395
5a+2b+13o+11p = 12.68

Row 4 - 5/21 · Row 1 → Row 4
21a+15b+8p = 7.57
2.8571b+2o+2.8571p = 2.6486
15.2857b+18o+13.619p = 18.0395
-1.5714b+13o+9.0952p = 10.8776

Pivot: Row 2 ↔ Row 3
21a+15b+8p = 7.57
15.2857b+18o+13.619p = 18.0395
2.8571b+2o+2.8571p = 2.6486
-1.5714b+13o+9.0952p = 10.8776

Row 3 - 2.85714286/15.28571429 · Row 2 → Row 3
21a+15b+8p = 7.57
15.2857b+18o+13.619p = 18.0395
-1.3645o+0.3115p = -0.7233
-1.5714b+13o+9.0952p = 10.8776

Row 4 - -1.57142857/15.28571429 · Row 2 → Row 4
21a+15b+8p = 7.57
15.2857b+18o+13.619p = 18.0395
-1.3645o+0.3115p = -0.7233
14.8505o+10.4953p = 12.7321

Pivot: Row 3 ↔ Row 4
21a+15b+8p = 7.57
15.2857b+18o+13.619p = 18.0395
14.8505o+10.4953p = 12.7321
-1.3645o+0.3115p = -0.7233

Row 4 - -1.36448598/14.85046729 · Row 3 → Row 4
21a+15b+8p = 7.57
15.2857b+18o+13.619p = 18.0395
14.8505o+10.4953p = 12.7321
1.2759p = 0.4465

p = 0.44654919/1.27585484 = 0.35
o = 12.73214953-10.495327102804p/14.85046729 = 12.73214953-10.4953271 · 0.35/14.85046729 = 0.61
b = 18.03952381-18o-13.619047619048p/15.28571429 = 18.03952381-18 · 0.61-13.61904762 · 0.35/15.28571429 = 0.15
a = 7.57-15b-8p/21 = 7.57-15 · 0.15-8 · 0.35/21 = 0.12

a = 3/25 = 0.12
b = 3/20 = 0.15
o = 61/100 = 0.61
p = 7/20 = 0.35

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