Johan has some boxes. He has only small boxes, medium boxes and large boxes. number of small boxes : number of medium boxes=2:5 number of medium boxes : number of large boxes=4:7 Johan has more than 150 boxes in total. What is the least possible number of large boxes that he could I have?

## Step1: Assess the Ratio<br />We interpret the given ratio for number of small boxes: number of medium boxes: number of large boxes.<br />Initially we are to assume \(2:5:7\), but from the text, it's clear that the number of medium boxes in both ratios aren't equal.<br /><br />## Step2: Equalize the Ratios<br />To match the number of medium boxes in both the ratios, we'll need to convert them into equivalent ratios. Therefore, we compute the least common multiple (LCM) of 5 and 4. <br />### \[LCM(5,4) = 20\]<br />Here, we adjust the 2 original ratios, as follows:<br /><br />\(\text{{Small boxes : Medium boxes}}\) ratio from \(2:5\) <br />Change it to: \(\frac{2}{5} \times \frac{20}{20} = 8: 20\)<br /><br />\(\text{{Medium boxes : Large boxes}}\) ratio from \(4:7\) <br />Change it to: \(\frac{4}{7} \times \frac{20}{20} = 20 : 35\)<br /><br />This leads us to a corresponding ratio of small: medium: large as \(8 : 20 : 35\).<br /><br />## Step 3: Total up the Ratio Parts<br />The ratio can be simplified further by adding all of them together to determine the total number of 'parts' each ratio represents.<br /><br />\(Total parts = 8 (small boxes) + 20 (medium boxes) + 35 (large boxes) = 63 parts.\)<br /><br />## Step 4: Division of Boxes<br />Johan has more than 150 boxes in total. And so, each part represents <br /><br />### \[\frac{150}{63}= 2.38 \]<br /><br />Being that a box can't be a fraction, we'll start from 3 boxes per part.<br /><br />## Step 5: Calculate Number of Large Boxes<br />We now determine the least number of large boxes present by multiplying the ratio for large boxes by 3 (number of boxes/ part),<br /><br />### \[3\times35 =105\]