Aditya sent us a very well explained solution to this problem:

Tim has $50$% blue, $35$% red, $10$% white and $5$% union jacks.

Beth has $40$% blue, $32$% red, $20$% white and $8$% union jacks.In fractions, this is:

T = $\frac{1}{2}$ blue, $\frac{7}{20}$red, $\frac{1}{10}$ white & $\frac{1}{20}$ union jack.

= $\frac{10}{20}$ blue, $\frac{7}{20}$ red, $\frac{2}{20}$ white and $\frac{1}{20}$ union jacks..

Therefore Tim has $20$ flags

B = $\frac{2}{5}$ blue, $\frac{8}{25}$ red, $\frac{1}{5}$ white & 2/25 $\frac{2}{25}$ union jacks.

= $\frac{10}{25}$ blue, $\frac{8}{25}$ red, $\frac{5}{25}$ white and $\frac{2}{25}$ union jacks..

Therefore Beth has $25$ flagsNow, we know that Beth has more flags than Tim. Beth has one more red flag, and both have the same number of blue flags. Between them, they have $3$ union jacks.

The second part of the problem:

Out of every $20$ flags Tim would have $1$ union jack.

Out of every $25$ flags Beth would have $2$ union jacks.

So first we thought about how many different ways you could
make $10$ union jacks

Tim | Beth |

8 | 2 |

6 | 4 |

4 | 6 |

2 | 8 |

For each of these we then need toknow the total number of flags :

Tim | Beth | Tim has this number of flags | Beth has this number of flags | Total number of flags |

8 | 2 | 160 | 25 | 185 |

6 | 4 | 120 | 50 | 170 |

4 | 6 | 80 | 75 | 155 |

2 | 8 | 40 | 100 | 140 |