Jane has a recipe to make butter cakes.
She also has $\dfrac{1}{2}$ kg of flour, 390 g of butter and 160 g of sugar.
What is the greatest number of pieces of butter cakes she can make?
(1) 20
(2) 25
(3) 32
(4) 39
(2)
$\dfrac{1}{2}\text{ kg} = 500\text{ g}$
Let one batch of ingredients make 10 pieces.
Flour:
$\dfrac{500\text{ g}}{200\text{ g/batch}} = 2.5 \text{ batches} \Rightarrow 2\text{ full batches}$
Butter:
$\dfrac{390\text{ g}}{100\text{ g/batch}} = 3.9\text{ batches} \Rightarrow 3\text{ full batches}$
Sugar:
$\dfrac{160\text{ g}}{50\text{ g/batch}} = 3.2\text{ batches} \Rightarrow 3\text{ full batches}$
Greatest number of full batches$$ \text{Minimum of } \{2, 3, 3\} = 2\text{ batches} $$
Greatest fraction of batches$$ \text{Minimum of } \{2.5, 3.9, 3.2\} = 2.5\text{ batches} $$
$$ 2.5 \times 10 = \boxed{25} $$
The answer is option (2).
Convert flour to grams:$$ \dfrac{1}{2}\text{ kg} = 500\text{ g} $$
Jaime has: $500\text{ g}$ flour, $420\text{ g}$ butter, $110\text{ g}$ sugar.
Recipe requires per 9 pieces: $150\text{ g}$ flour, $100\text{ g}$ butter, $50\text{ g}$ sugar.
Check if 27 pieces (3 batches) can be made:
Flour needed: $3 \times 150\text{ g} = 450\text{ g}$ (Have $500\text{ g}$)
Butter needed: $3 \times 100\text{ g} = 300\text{ g}$ (Have $420\text{ g}$)
Sugar needed: $3 \times 50\text{ g} = 150\text{ g}$ (Have $110\text{ g}$)
Sugar is insufficient for 3 batches. Thus, 27 pieces cannot be made.
Since 27 pieces cannot be made, any greater number of pieces (36 or 81) also cannot be made.
The greatest number of pieces Jaime can make is 18.
The answer is option (1).$$\boxed{18}$$
