Liu wants to decorate some cakes with shapes. She has 140 shapes. Each shape is a star or a heart. The ratio of the number of stars : number of hearts is 4 : 3. She wants to put 5 stars and 4 hearts on each cake. How many cakes can Liu decorate? Show full working to support your answer.

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Liu wants to decorate some cakes with shapes - OCR - GCSE Maths - Question 9 - 2018 - Paper 1

Question 9

Liu wants to decorate some cakes with shapes. She has 140 shapes. Each shape is a star or a heart. The ratio of the number of stars : number of hearts is 4 : 3. She... show full transcript

Worked Solution & Example Answer:Liu wants to decorate some cakes with shapes - OCR - GCSE Maths - Question 9 - 2018 - Paper 1

Step 1

Determine the number of stars and hearts

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Answer

Given the ratio of stars to hearts is 4 : 3, we can denote the number of stars as 4x and the number of hearts as 3x. Together, they sum to the total number of shapes:

$4x + 3x = 140$

Simplifying gives:

$7x = 140$

Dividing both sides by 7 yields:

$x = 20$

Thus, the number of stars is $4x = 80$ , and the number of hearts is $3x = 60$ .

Step 2

Calculate the number of cakes Liu can decorate

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Answer

Liu wants to use 5 stars and 4 hearts for each cake. Therefore, the total number of shapes needed per cake is:

$5 + 4 = 9\text{ shapes}$

Now we can calculate the total number of cakes she can decorate by dividing the total number of shapes by the shapes needed per cake:

$\frac{140}{9} \approx 15.56$

Since she can't decorate a fraction of a cake, we round down to 15. Thus, Liu can decorate 15 cakes.

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