multiplicative-ratio-constraints
Quantities are linked by ratios (R = G/2, B = 2G, or a:b given). Combined with the requirement that all counts are whole numbers, the ratios force totals to lie in a specific arithmetic progression (multiples of some k). The unique answer is the choice that fits the multiplicative scheme.
How to solve
- Change ratio_chain from 1:2:4 to 2:3:5 — totals must now be multiples of 10 instead of 7
- Add a perturbation that changes the ratio from 3:1 to 5:2 (problem 21 style)
- Convert from 'which is possible' to 'find the smallest total possible'
Sub-archetype mix (6)
Click a row to see member problems.
- direct-ratio-scale-up 24% (32)
A single a:b ratio is stated, along with one known value or a total constraint; find the unknown quantity by writing each part as a multiple of k and solving for k.
no figureamc10-2004A-13no figureamc10-2004B-7no figureamc10-2005A-11no figureamc10-2006A-3no figureamc10-2007A-4
amc10-2012B-2no figureamc10-2016A-3no figureamc10-2016B-11no figureamc10-2016B-7no figureamc10-2021A-2no figureamc12-2002A-4no figureamc12-2002B-9no figureamc12-2004B-5no figureamc12-2005A-10no figureamc12-2006A-3no figureamc12-2007A-3no figureamc12-2011B-6no figureamc12-2012A-3 amc12-2012B-2no figureamc12-2014A-2no figureamc12-2016B-4no figureamc12-2016B-9no figureamc12-2024B-21no figureamc8-2009-15no figureamc8-2009-2no figureamc8-2010-11no figureamc8-2014-7no figureamc8-2016-16
amc8-2017-2no figureamc8-2017-6no figureamc8-2022-6no figureamc8-2023-5 - chained-ratio-linking 16% (21)
Two or more ratios share a common variable; standardize that shared term via LCM or substitution to express all quantities in a single unified ratio, then answer a count or minimum-total question.
no figureamc10-2005B-16no figureamc10-2007B-15no figureamc10-2008A-12no figureamc10-2008B-6no figureamc10-2015A-4no figureamc10-2016B-13no figureamc10-2019B-11no figureamc10-2020A-9no figureamc10-2020B-3no figureamc12-2005B-12no figureamc12-2006A-7no figureamc12-2007B-11no figureamc12-2007B-4no figureamc12-2008B-8no figureamc12-2012A-5no figureamc12-2019A-19no figureamc12-2020B-3no figureamc12-2022A-2no figureamc8-1999-22no figureamc8-2008-20no figureamc8-2013-16 - percent-fraction-recovery 22% (29)
A fraction or percentage of one group equals a known count or another group's fraction; recover the whole population size, or compute what fraction of the combined total one subgroup represents.
no figureamc10-2004B-14no figureamc10-2005A-1no figureamc10-2009A-7no figureamc10-2011A-8no figureamc10-2012A-7no figureamc10-2012B-5no figureamc10-2013A-10no figureamc10-2017A-14no figureamc10-2018A-2no figureamc10-2018A-6no figureamc10-2021B-3no figureamc10-2021B-6no figureamc10-2025A-2no figureamc12-2005A-1no figureamc12-2011A-5no figureamc12-2012A-4no figureamc12-2013A-3no figureamc12-2019A-1no figureamc12-2019A-2no figureamc12-2021B-4no figureamc12-2025A-2no figureamc8-1999-12no figureamc8-2002-24no figureamc8-2004-20no figureamc8-2004-22no figureamc8-2010-13no figureamc8-2011-14no figureamc8-2015-16no figureamc8-2016-12 - integrality-divisibility-screen 13% (18)
Ratio fractions applied to an unknown total T require T to be a multiple of the LCM of the denominators; combined with a bounding inequality, this divisibility constraint identifies the unique valid answer among the choices.
no figureamc10-2002B-7no figureamc10-2016A-5no figureamc10-2018B-14no figureamc10-2024A-19
amc10-2024B-25no figureamc12-2002B-15no figureamc12-2002B-4no figureamc12-2004B-22no figureamc12-2017A-11no figureamc12-2018B-10no figureamc12-2018B-17no figureamc12-2024A-12no figureamc8-2002-25no figureamc8-2017-12no figureamc8-2017-9no figureamc8-2019-23no figureamc8-2023-22no figureamc8-2024-9 - ratio-shift-two-state 12% (16)
An initial ratio between two groups is given, then a fixed number of items are added to or transferred between groups, yielding a new ratio; set up a system of two ratio equations to recover the original or final counts.
no figureamc10-2002A-17no figureamc10-2007B-14no figureamc10-2010A-3no figureamc10-2017B-5no figureamc12-2002A-10no figureamc12-2005B-21no figureamc12-2007B-10no figureamc12-2014B-6no figureamc12-2015B-5no figureamc12-2018A-1no figureamc12-2020B-5no figureamc12-2021A-6no figureamc8-2007-20no figureamc8-2020-13no figureamc8-2024-21no figuresm-10 - multiplicative-rate-or-series 13% (18)
A constant multiplicative factor (a per-unit rate, a similarity scale, or a per-step fraction) must be identified from given data and then either compared across options or applied repeatedly as a geometric series to find a total.
no figureamc10-2002B-21no figureamc10-2010A-12no figureamc10-2014B-6
amc10-2015A-14no figureamc10-2015A-9no figureamc10-2016B-10
amc10-2024A-16no figureamc12-2002B-17no figureamc12-2008A-8no figureamc12-2008B-14no figureamc12-2010A-11no figureamc12-2010A-7no figureamc12-2014B-2no figureamc12-2015A-7no figureamc12-2016B-8no figureamc12-2025A-7no figureamc8-2005-22no figureamc8-2012-13
More data (year-over-year, tool fingerprint, grade distribution, all members)
Tool fingerprint (1–17)
Grade distribution
- Gr 3 1
- Gr 4 8
- Gr 5 2
- Gr 6 45
- Gr 7 13
- Gr 8 10