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Question

Difficulty: MediumRatios and Proportions

A sporting goods store stocks basketballs and footballs in a ratio of 3:43:4. After receiving a new shipment of 1010 additional basketballs, the ratio of basketballs to footballs changes to 5:65:6. How many footballs are currently in stock?

  1. A
    20
  2. 120Answer
  3. C
    100
  4. D
    12

Answer

There are 120 footballs currently in stock.
By scaling the initial ratio (3:43:4) to 9:129:12 and the new ratio (5:65:6) to 10:1210:12, we can see that the 1010 added basketballs represent exactly 11 part (10−9=110 - 9 = 1). Since the footballs represent 1212 parts in this scaled system, there are 12×10=12012 \times 10 = 120 footballs in stock.

Step-by-Step Solution

1
Identify the quantity that remains unchanged.
The number of footballs does not change when the new shipment of basketballs arrives.
This provides an anchor point to compare the before and after ratios.
2
Find a common multiple for the football ratio parts.
The initial ratio has 44 parts for footballs and the new ratio has 66 parts. The least common multiple (LCM) of 44 and 66 is 1212.
Creating a common denominator for the unchanging item allows direct comparison of the changed item's parts.
3
Scale both ratios so the football parts equal 1212.
Initial ratio: Multiply 3:43:4 by 33 to get 9:129:12. New ratio: Multiply 5:65:6 by 22 to get 10:1210:12.
This standardizes the 'size' of one part across both ratios.
4
Determine the change in basketball parts.
The scaled basketball parts increased from 99 to 1010, a difference of exactly 11 part.
This 11-part increase corresponds directly to the physical items added.
5
Calculate the actual number of footballs.
Since 11 part equals the 1010 basketballs added, the 1212 parts for footballs equal 12×10=12012 \times 10 = 120 footballs.
This yields the final answer by multiplying the scaled parts by the unit value.

Key Concept

Scaling Ratios and Proportions

Alternative Method

Using algebra, let the initial basketballs be 3x3x and footballs be 4x4x. After adding 1010 basketballs, the new ratio yields the equation 3x+104x=56\frac{3x + 10}{4x} = \frac{5}{6}. Cross-multiplying gives 6(3x+10)=5(4x)6(3x + 10) = 5(4x), which simplifies to 18x+60=20x18x + 60 = 20x. Solving for xx gives 2x=602x = 60, so x=30x = 30. The number of footballs is 4x=4(30)=1204x = 4(30) = 120.
Estimated Time:1m 0s
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Topics

Ratios and Proportions

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