Let the number of pages Danny read on Monday be \( d \). Since Ian read four times as many pages as Danny on Monday, he read \( 4d \) pages.
On Tuesday, Danny read 54 pages. So, by the end of Tuesday, the total number of pages read by each person can be expressed as follows:
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Pages read by Danny by the end of Tuesday: \[ d + 54 \]
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Pages read by Ian by the end of Tuesday: \[ 4d \]
According to the problem, by the end of Tuesday, Danny and Ian were on the same page. Therefore, we can set up the equation: \[ d + 54 = 4d \]
To solve for \( d \), we first rearrange the equation: \[ 54 = 4d - d \] \[ 54 = 3d \]
Now, divide both sides by 3: \[ d = \frac{54}{3} = 18 \]
Thus, Danny read 18 pages on Monday.
To verify, if Danny read 18 pages, then Ian read \( 4d = 4 \times 18 = 72 \) pages on Monday. By the end of Tuesday, Danny's total pages read would be: \[ 18 + 54 = 72 \] Indeed, Danny and Ian both read 72 pages by the end of Tuesday, confirming our solution is correct.
Therefore, the number of pages Danny read on Monday is \( \boxed{18} \).
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