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There are some small cubes and some large cubes in a bag - Edexcel - GCSE Maths - Question 18 - 2019 - Paper 2
Question 18
There are some small cubes and some large cubes in a bag. The cubes are red or the cubes are yellow. The ratio of the number of small cubes to the number of large cu... show full transcript
Worked Solution & Example Answer:There are some small cubes and some large cubes in a bag - Edexcel - GCSE Maths - Question 18 - 2019 - Paper 2
Step 1
Explain why the least possible number of cubes in the bag is 88
Answer
To find the least number of cubes in the bag, we must determine a common multiple of the ratios provided.
The ratio of small cubes to large cubes is 4:7, which means for every 11 cubes (4 small + 7 large), we can express the total number of cubes as a multiple of 11.
The least common multiple (LCM) of 4 and 7 is 28, and to find the minimum number of cubes, we can multiply by 4 (the smallest integer) to maintain the ratio, giving us
y = 4k + 7k = 11k,
where k must be an integer that satisfies the condition for both ratios.
When k = 8, we have 88 cubes, since
y = 11 × 8 = 88.
Thus, the least possible number of cubes in the bag is 88.
Step 2
Work out the least possible number of large yellow cubes in the bag
Answer
Since all the small cubes are yellow, we know that all the small cubes will contribute to the yellow count.
Let the number of small cubes be 4k and the number of large cubes be 7k. Since k = 8, we have
- Number of small cubes = 4 × 8 = 32
- Number of large cubes = 7 × 8 = 56
Next, finding the ratio of red cubes to yellow cubes (3:5), the number of red cubes would be
rac{3}{8} imes ext{Total Cubes} = rac{3}{8} imes 88 = 33.
Thus, number of yellow cubes = Total Cubes - Red Cubes = 88 - 33 = 55.
Given the ratio of large cubes to yellow cubes (5 parts yellow out of 8), we can conclude that the least possible number of large yellow cubes:
Since 55 yellow cubes include both small and large and we know all small cubes are yellow,
so large yellow cubes would be
ext{Total number of yellow cubes} - ext{Number of small cubes} = 55 - 32 = 23.
Therefore, the least possible number of large yellow cubes in the bag is 23.