Julibella's Math Question

Chapter 2.3
22) Taj has three scoops for measuring flour. The largest scoop holds 2 1/2 times as much as the smallest one. The middle scoop holds 1 3/4 times as much as the smallest one.
Describe two different ways in which Taj could measure each of the following quantities. He can use full scoops only.
a) 3 1/4 times as much as the smallest scoop holds
b) 1/2 as much as the smallest scoop holds

First you would have to find the smallest scoop. You think to yourself, " the largest scoop (2 1/2) multiplied by what gives you the largest scoop (2 1/2)?" Then you think to yourself, "the middle scoop (1 3/4) multiplied by what gives you the middle scoop (1 3/4)?"

2 1/2 x _?_= 2 1/2
2 1/2 x _1_= 2 1/2

1 3/4 x _?_= 1 3/4
1 3/4 x _1_= 1 3/4

**What we know**

Can only measure with full scoops.
Smallest scoop equals = 1
Middle scoop= 1 3/4
Largest scoop= 2 1/2

a) One way: 3 1/4 = three middle scoops, take away one small scoop
3 1/4 = 3 x 1 3/4 - 1
3 1/4 = 3/1 x 7/4 - 1
3 1/4 = 21/4 - 1
3 1/4 = 4 1/4 -1
3 1/4 = 3 1/4

Second way: 3 1/4 = one large scoop, plus one middle scoop, take away one small scoop
3 1/4 = 2 1/2 + 1 3/4 - 1
3 1/4 = 5/2 + 7/4 - 1
3 1/4 = 10/4 + 7/4 -1
3 1/4 = 17/4 - 1
3 1/4 = 4 1/4 -1
3 1/4 = 3 1/4

b) One way: 1/2 = one large scoop, take away two small scoops
1/2 = 2 1/2 - 2 x 1
1/2 = 2 1/2 - 2
1/2 = 1/2

Second way: 1/2 = two middle scoops, take away three small scoops
1/2 = 2 x 1 3/4 - 3 x 1
1/2 = 2/1 x 7/4 - 3 x 1
1/2 = 14/4 - 3 x 1
1/2 = 3 2/4 -3
1/2 = 2/4
1/2 = 1/2