Three tankers contain $403$ litres, $434$ litres and $465$ litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers an exact number of times. Also find the total number of times the measuring container will be used.
Step-by-Step Solution
Key Concept: Maximum capacity $= \text{HCF}(403, 434, 465)$. Total measurements $=$ sum of volumes / HCF.
Capacity of measuring container $= \text{HCF}(403, 434, 465)$. [0.5 Mark]
Prime factorisation:
$403 = 13 \times 31$
$434 = 2 \times 7 \times 31$
$465 = 3 \times 5 \times 31$. [2.0 Marks]
$\text{HCF}(403, 434, 465) = 31$ litres. Maximum capacity of container $= 31$ litres. [1.0 Mark]
Number of times container used for 1st tanker $= 403 / 31 = 13$.
For 2nd tanker $= 434 / 31 = 14$.
For 3rd tanker $= 465 / 31 = 15$. [1.0 Mark]
Total number of times container is used $= 13 + 14 + 15 = 42$ times. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Identifying HCF requirement: 0.5 Mark
Prime factorisation of 403, 434, 465: 2.0 Marks
Evaluating HCF $= 31$ litres: 1.0 Mark
Calculating usage count for each tanker (13, 14, 15): 1.0 Mark
Total usage count $= 42$ times: 0.5 Mark
Correct Answer: